Fuzzy Reasoning in Information, Decision and Control Systems (Intelligent Systems, Control and Automation: Science and Engineering)

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) \iy<.z 3. 'f{xX{y,zy)-'f{'f{x,y)^) 4. T(x,0) = 0 Definition 5.3 The function N;[0,l]-*[0,1] is called negation function if and only if N possesses the following properties for all x,ye[0,l]: 1. N(0)=1 , N(1) = 0 2. N{x)
TfA^,y) min{x,y)

2.

x-y

3.

max(x+y-l , 0)

4.

x-y x + y-x-y

5.

1 - ( 1 - A ) - ( X + 7 - A : •y)

\-x-y

6.

xy max{x,y,X)

T^'i^.y)

N(x)

Researcher

max{x,y)

l-x

Zadeh

1-x

Gogeuen

l-x

Giles

\-x

Bandler

l-x

Hamaster

l-x

Dubois

x+y-x-y mm{x+y,

1)

x+y-2-x-y 1 -x-y A-(x + y)+Ar-7-(l-2-A) A + ;r-y(l-A) j_

(\-x)-il-y) max(l --V, l - y , A )

Actually, not any single triple of T-operators is the best in all decision making problems. The simplest triple is the one suggested by Zadeh, but it is not the most suitable in many cases. For a detailed discussion of T-operators the reader is referred to [15]. 6. THE FUZZY SETS AS HYPERCUBES 6.1. The Hypercube Geometric Representation The definition of fuzzy sets as sets of pairs and their representations as curves possess some drawbacks with re.spect to their utilization. An alternative easy to use geometric

13 representation is based on the concept of hypercube, i.e. fuzzy sets are represented as points in hypercubes.

Definition 6.1 Let A = [ (x,fi^(x))

( xeX , p^(jc):>4-[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(*<)K

16 For p=2 the distance L\A,B) is the fuzzy Euclidean distance between A and B, and for p= 1 the distance 0(A,B) is the fuzzy Hamming distance between A and B. In the special case where A is a fuzzy set and B=« (the empty set), then L'(^.B=0) = "£ |P^(x,.)-M,(x,)| (.1

(=1

i'l

(=1

= \A\ Now, let us consider the question of measuring the fuzziness of a fuzzy set. To this end, the entropy concept will be used and the problem will be dealt with geometrically. Definition 6.3 Consider the fuzzy set in the hypercube L" = [0,1]" and let a = L^(AA„eaT) ^ 'he Hammingdistance of A from the nearest vertex iA„g^), and b=L^(AyAf^j.) be the Hamming distance of A from the farest vertex (Af^j). Then as fuzzy entropy of A we define the quantity E(A)

The quantities a and b are geometrically shown in Fig. 12. (X2i=(0,l)

{x^jt^=(l,l)

(

,,_

• fuzzy set A A=((x,,l/3),(x ^/4))

II

1

^1

|x,|=(l,0) Fig. 12. Geometric representation of L'(A,Apgj,f) and L^(A,Aj-3j). O=(0,0)

Theorem 6.1 The fuzzy entropy E{A) of the fuzzy set A is equal to

EiA) - l^^-^^l IAUA^I

17

Proof The proof will be given with the aid of the hypercube concepts. For simplicity we use n=2 (Fig. 13). By definition E(A) = a/b. Due to the symmetric position of the fuzzy sets A, A'^, AnA^, AuA'^ one can see that a = a'and b=b', i.e. a'= |>\rL4'^| and b'= \AuA'^\. Thus J5(A) = £ = i ! = b

hi

l^^^'^l \A\JA'

Remark: \{ A is a crisp set then obviously Ar\A'^=0, i.e. |>4n4'^|=0 and |>\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 <£, and (ii) a set of morphisms from A to B, for every two objects A,Be^ (symbolically (it{A,B) ). The object of class ® may be sets, groups, topological spaces and so on. In the special case where the objects are sets, the category consists of; (i) the class ® of sets, and (ii) the set of aU possible functions from A to B (symbolically &iA,B) or f:A-*B) for every two sets A,Be^. The following table shows a sample of objects and their respective morphisms and structures. Objects

Morphisms

Structure

Sets

Functions

Points

Groups

Homomorphisms

Multiplication

Vector Spaces

Linear mappings

Linearity

Topological Spaces

Continuous functions

Nearness

Metric spaces

Metric functions

Distance

Fuzzy sets

?

7

8.2. Lattices The interval [0,1] and its properties plays a central role in the concept of fuzzy set. Definition 8.1 Lattice is defined to be a partially ordered set L that is associated with an ordering operator possessing the properties of reflexivity, antisymmetricity and transitivity and for which for every two elements a,beL there exist two new elements (aAb), (aVb) eL, which are called minimum and maximum respectively, such that a/\b<.a, b^aVb. The set [0,1] is a lattice according to the above definition under the ordering operation <. and the usual min and max concepts. Definition 8.2 Let L be a lattice and X a crisp set. Then as L fuzzy set we define the set A = {(Ar,^^(x)) \xeX,fi^{x):A^L

}

Clearly, the L-fuzzy set is a generalization of the standard fuzzy set where the lattice i s L = [0,l].

21

Definition 8.3 [17] The category set (L) consists of (i) objects {X,U) where X is a crisp set and U:X-*L in a fuzzy subset of X on the lattice structure, (ii) a morphism-function f:X-*Ysuch that v°/iu. At this point a generalization of the fuzzy set concept at higher level will be provided [18]. Consider a fuzzy subset of X, namely

A = { (x,p^(x)) I x€X , ^iAix):A^[0,l]

}

The main issue is that the fuzzy set is fully characterized by the membership function [ipfiC) which for every xeX returns a number. An obvious generalization would be to assume that fi^(x) has the form ^^(x):X-*S where S= {set of sets}, i.e. the standard //^(x) returns a number from the point set [0,1] whereas the generalization /i^(x) returns a set from S. Definition 8.4 The set A is called C-fuzzy set (or type-2 fuzzy set) if and only if A = { (x,ju^(x)) I xeX , p^(x):A-^S={set of sets} } Example 8.1 We are given that: "The number 3 is small with degree of membership 'about 0.8'". Here there is uncertainty not only on the elements Xj but also on the degrees of membership ^^(A-y), because the statement 'about 0.8' is a fuzzy set. 9. FUZZY REASONING 9.1. Linguistic Variables From the previous discussion one can see that the adherence in the extension of classical mathematical concepts (like the number, the matrix, the determinant, etc.) to fuzzy logic faces many difficulties and in some cases makes the situation more complex than what actually is. Thus to overcome these difficulties one has to resort to knew concepts. A new concept which plays a basic and dominant role in fuzzy logic and reasoning is the concept of linguistic variable. Zadeh justifies the introduction of linguistic variables as follows: "... The motivation for the use of words or sentences rather than numbers is that linguistic characterizations are, in generaljess specific than numerical ones ..." Definition 9.1 Linguistic variable is a variable the values of which are not numbers but words or sentences or propositions in a natural or artificial language. More formally the linguistic variable is defined as follows.

22

Definition 9.2 Linguistic variable is the quintuple <x,T(x),U,G,M> where x is the name of the linguistic variable, T(x) is the set of its values {term set), U is the reference hyperset upon which T(x) is structured (recall that the values of a linguistic variable are fuzzy sets), G is a syntactic rule that generates the names x, and M is a semantic rule that gives sense (meaning) to the names. Example 9.1 Consider the linguistic variable X-"age" with T(x) = { very young, not very young, pretty young, a little old, pretty old, not very old, very old }. Then a meaning of the fuzzy set "old" is M„,d={(u, t x „ , » ) , ue[0,100]}, where

f^ou("^ =

1+

^r

,

«e[0,50]

,

«e[50.100]

The linguistic variable "age" and its values are pictoriaUy depicted in fig. 14. linguistic variable

values of the linguistic varial

15 20 25

30

35

40

45

50

55

60

65

70

75

80

reference superset Fig. 14. Linguistic variable "age" and its "linguistic values".

Definition

9.3

A linguistic variable x is said to be structured if its set of values T^x) and the set of meanings M(x) that are given to them can be determined algorithmically. Example 9.2 Consider the linguistic variable "truth" with the following set of values: T(truth) = {true, not true, very true, not very true, false, not false, very false, not very false}. It is clear that TJx) can be determined algorithmically, and so the variable "truth" is a structured variable.

23

9.2. Linguistic Modifiers Linguistic modifier (hedge) is an operator which when applied to a value (fuzzy set) of a linguistic variable changes the meaning of the value. For example, if .4 is the term set, then when the linguistic modifier m is applied to the elements of A, it produces the new term set B=m(A). Some linguistic modifiers are the following: (i) f^con(A)(") = (P^("))^ Concentration (ii) ^^dil(A)(") = (^Ai")y' Dilation (iii)

f'uu.J'*)

[ 2(/i^(«))2 - \ , ,,, , ,,, [1-2(1-//^(K))^

, ,

/i^(tt)e[0.0.5] , , ,„, „ ii^(,u) e[0.5,l]

Contrast Intensification

Example 9.3 Let A be an element of the term set of a linguistic variable. Clearly, A is fuzzy set. Then the most common linguistic modifiers applied upon A are: i) very(A) = con(>\) ii) more or less (^) = dil(yi) iii) slightly(^) = Int[(y4) and not (very (A))] The grammatical correspondence of linguistic modifiers are the adverbs and some adjectives. Classical examples are the following: very, very much, pretty, nearly pretty, not at all, almost not at all, etc.

9.3. Generalized Modus Ponens It is well known taht in the classical logic the proof process is based entirely on the "logical tautology" concept. The main classical tautologies are: Modus Ponens: {A/\(A->B)}-^B Modus ToUens: {{A^B)h-B)^A Chain syllogism: {(A-»B)A(5-»C)}-(A-C) Logic Inversion: {A-'B)-*(~B-^~A) Any other tautology can be reduced to one of the above. In this section we shaU generalize the modus ponens (MP) rule such that to be applicable in an uncertain environment. To this end, let us start with an illustrative example. The MP rule is written in the following general form: Fact: A Implication: IF A THEN B Conlclusion: B For example Fact: Tomato red Implication: IF (Tomato red) THEN (Tomato mature) Conclusion: Tomato mature We observe that, in order for the rule to apply and draw a conclusion, the fact nust coincide with the antecedent (left hand side) of the implication.

24

Let us now permit the fact not to coincide with the antecedent of the implication, but to be "alike". For example Fact: Tomato very red Implication: IF (Tomato red) THEN (Tomato mature) Conclusion: ? Of course a conclusion can be drawn by a strict application of the MP rule, but it is a real life conclusion that now the tomato is very mature, something which does not strictly coincide with the conclusion, but it is "alike" the MP conclusion. On the basis of the above observation one can state a Generalized Modus Ponens (GMP) rule as: Fact: A' Implication: IF A THEN B Conclusion: B' where now A', A, B' and B are fuzzy sets. It is true that the GMP rule mimics much better the human type of reasoning than the classical MP rule, since it can work not only under idealy identical conditions but also under general conditions of similarity (alikeness). This is very useful! since reduces very much the number of rules that are required to have in the knowledge base of the application. 9.4. The Max-Min Compositional Inference Rule The max-min compositional rule was introduced by Zadeh in 1973 as a way of reasoning under uncertainty [19]. Definition 9.4 Let A be a fuzzy set on X, B a fuzzy set on Y, and R a fuzzy relation on XxY, i.e. A = { (x,fiAix)

I xeX , ^ ^ ( x ) : A - [ 0 , l ] }

B= { {y,^B(y)

|yey,^B(y):5-[o,i] }

R = { ((^.y).M/?(^.j)) I ix,y)€X€Y

, p^(x,y) = min{^^(x),^£(y)} }

Then, given A and i?, B is found as B=A»/?, /^^(y) = max{min{/i^(x),jU^(x,y)} } where "»" is the symbol of the max-min composition operator. Example 9.4 Consider the foUowing: (a) X={1,2,3,4}, (b) A = "x smaU"={(l,l), (2,0.6), (3,0.2), (4,0)}, (c) R = "x nearly equal to y with matrix

25 R

y

X

1

2

3

4

1

1

0.5

0

0

2

0.5

1

0.5

0

3

0

0.5

1

0.5

4

0

0

0.5

1

It is desised to find what happens with the variable y, i.e. to determine the fuzzy set B="y'. To this end, use of the max-min composition rule of inference is made, i.e. B=A°R, where M B ( 7 ) = msx{min{^A{x),Hji{x,y)}

} = { ( 1 , 1 ) , (2,0.6), (3,0.5), (4,0.2) } .

X

Thus, IF "x smaU" and "x nearly equal to y THEN "y nearly small".

9.5. Estimation of the Membership Function: Defuzzification The estimation of the membership function which leads to defuzzification can be done in several ways that are arbitrary. The membership function /U^(jf) that is mostly used is the triangular one which possesses a maximum value (usually 1) in the most representative and characteristic value of the fuzzy variable x. Thus in this case the defuzzification leads to the value of maximum Five other more systematic methods of defuzzification (estimation of /i^(x)) are the following. (j) Average guess method In this method we calculate the average value of the values of /i^(;if) that are suggested by experts. This is used as the best estimate of H^ix). The corresponding value of x is the required deffuzified value of x. (ii) Distance function method Initially we compute a distance d{x).oi the arbitrary point x from the fuzzy set A under consideration. For the elements that belong to A this distance is obviously zero. For those X not belonging to A this distance takes some maximum value say supd. On the basis of the above the membership value is given by PA(^)

= 1

d(£) supd

(Hi) Intuitive relation method It is intuitively true that the rate of change of /u^(^) must increase when the belief that x belongs to the fuzzy set A is strengthened. Analytically this is expressed by the relation

26

d/i^(x)

X/z^(x)[l-/i^(x)] dx which upon integration gives the membership function 1 + exp(a-bx) The constants a and b are determined by the other data of the problem. (iv) Binary polling method In this method we form a group of experts and ask them whether "x belongs to the set A'. Their reply should be binary, i.e. yes or no. Then the value of iu^(x) is estimated by: , V _ number of positive replies yes total number of replies (v) Relative preference method Let Abe Si discrete fuzzy set A = { {x,^iJ^{x)) I x e X , ix^{x) : ^ - [ 0 , 1 ] } As relative preference otx^ with respect to X: is defined the quantity _ MA(^I)

We now form the matrix P=[Py] which possesses the following three properties: •P«=landp^.= l / p . . • All eigenvalues of P are zero except one which is equal to the cardinality n of A. • The eigenvector w that corresponds to the nonzero eigenvalue n has as its elements the values of the membership function corresponding to the elements of A, i.e. w=\ii^{x;)\ = \ii^{x{) ... iiA{x„)VOf course, this method is applicable to discrete finite fuzzy sets, not to continuous fuzzy sets that have an infinite number of elements. 10. A LOOK AT FUZZY OR LINGUISTIC CONTROL Here a brief description of the structure of the basic fuzzy control loop will be provided. Fuzzy control uses linguistic variables (this is why it is sometimes called linguistic control) and mimics the human action more closely than traditional control. For comparison, Fig. 15 depicts the architecture of both the traditional and the basic fuzzy control loop. The problem in the traditional control loop is to design a controller which will accept the error e=r-y as input and will give an output (control signal) u such that the output y of the overall system follows the set point (reference input) r as nearer as possible.

27

r

^

Controller

Process

(a)

Rule Base

inference mechanism

(b)

terms

terms

fuzziflcation

defuzziflcation

Z^

Process

Fig. 15. (a) Traditional control loop (b) Basic fuzzy control loop The problem in the fuzzy control loop is to design an inference mechanism (fuzzy controller) which will mimic the human type of syllogism and will ensure a desired performance of the overall system. The fuzzy control loop involves the following blocks: a rule base, an inference mechanism and the fuzzification/defuzzification blocks. The fuzzification/defuzzification blocks have been discussed earlier. The fuzzy knowledge base contains the rules that are to be used for the control of the process. These rules are usually the result of interviews with the expert operators (very rarely come out of mathematical analysis or simulations) and have the form IF-THEN. In the general case the rules have many inputs and many outputs (MIMO). However, it can be shown [20] that a set of MIMO rules can be transformed to a set of MISO (multiple input - single output) rules. As an example here we consider rules with two inputs and one output, i.e. rules of the form Rji IF "x is A{ and "y is B{ THEN "z is C{ A rule of this type has two assumptions (premises) 'x is A{ and "y is B{ which are defined on the cartesian product t/xV. The whole rule constitutes a fuzzy relation in the fuzzy Cartesian product [/xVxW, i.e. fi^^ = /'(^^^a, nwi c,/"'^-^)

where U, Vand

W are the respective fuzzy sets. An easy to use method for the calculation of the relation R makes use of the min operator, namely

28 t^R^ = f^(A^ and B,- then q)^"'''''*') = niin{/\y(u),B,.(v),C;(lv)} If the knowledge base contains a total of n rules (Rj , /=l,2,...,n) then it can be regarded as a unique relation R where R=

U Rj

Suppose now that at a certain instant of time we have observed in the process that 'x is A" and "/ is B". The problem is to combine this fact with the rules of the knowledge base such that to produce a suitable control y. To this end one must use the max-min composition scheme, i.e. n (A',B')O

n

U /?,- = U j=l

(A',B^)'>RI

/=I

The right hand side of the above identity suggests that instead of applying the fact (A',B') to the knowledge base

n U Rj

as a whole, one can apply {A',B') to each rule Rj of the

1=1

knowledge base separately. The proof of the validity of the above identity can be found in [20]. In this way, for each input, one can evaluate the individual contribution of each rule to the final result. This facilitates very much the control process, reduces considerably the computational load and helps to have a better monitoring which rules are fired and how much are fired. 11. CONCLUSIONS Scientists and engineers recognise the fact that "fuzziness" is inherent everywhere in the real human life. This is particularly true whenever the human factor has a direct and active effect (social issues, economical issues etc.) Actually, the effort to solve uncertain problems by classical (deterministic) mathematical theory may lead to "deadends". In general, mechanistic type systems can be treated by quantitative analysis, while humanistic type .systems are too fuzzy to be amenable to such exact mathematical methodologies. An elegant approach to handle such uncertain situations is the fuzzy set theory which has now arrived at a very mature state of expansion and application. In this chapter an effort was made to give within a strictly limited size an appreciation of fuzzy set and fuzzy logic theory. Further concepts and developments can be found in the references and in the rest of this book. REFERENCES 1. L.A. Zadeh: Fuzzy Sets, Inform, and Control, vol.8, pp.338-353 (1965) 2. L.A. Zadeh: Fuzzy Algorithms, Inform, and Control, vol.11, pp.323-339 (1969)

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29 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

S.S.L. Chang and L.A. Zadeh: On Fuzzy mapping and Control, IEEE Trans. Sys. Man Cybern., vol. SMC-2, pp.30-34 (1972) L.A. Zadeh: Outline of a New Approach to the Analysis of Complex Systems and Decision Processes, IEEE Trans. Sys. Man. Cybern., vol. SMC-3, pp.28-44 (1973) L.A. Zadeh: A Computational Approach to Fuzzy Quantifiers in Natural Languages, Comp. & Maths., vol.9, pp. 149-184 (1983) L.A. Zadeh: Fuzzy Logic and Approximate Reasoning (In Memory of Grigore Moisil), Synthase, v.30, pp.407-428 (1975) L.A. Zadeh: Commonsense Knowledge Representation Based on Fuzzy Logic, Computer, pp.63-65, Oct. 1983. R.M. Tong: A Control Engineering Review of Fuzzy Systems, Automatica, vol. 13, pp.559-569 (1977) P.M. Larsen: Industrial Applications of Fuzzy Logic Control, Int. J. ManMachine Studies, vol.12, pp.3-10 (1980) M. Black: Reasoningwith Loose Concepts, Dialogue, vol. 2, pp.1-12 (1963) R. Bellman and M. Giertz: On the analytic Formalism of the Theory of Fuzzy Sets, Information Science, vol.5, pp.149-156 (1973) B. Gain: Precise Past - Fuzzy Future, Intl. J. Man-Machine Studies, vol.19, pp. 117-134 (1983) S. Haack: Do we Need Fuzzy Logic?, Intl. J. Man-Machine Studies, vol.11, pp.437-445 (1979) R.M. Tong: Analysis and Control of Fuzzy Systems Using Finite Discrete Relations, Int. J. Control, vol.32, (1977) M.M. Gupta: Theory of T-norms and Fuzzy Inference Methods, Fuzzy Sets and Systems, vol.40, pp.431-450 (1991) B. Kosko: Neural Networks and Fuzzy Systems: A Dynamical Systems Approach to Machine Intelligence, Prentice Hall, 1992. C.V. Negoita and D.A. Ralescu: Simulation, Knowledge-Based Computing and Fuzzy Statistics, Van Nostrand Reinhold, 1987. C.V. Negoita and D. Ralescu: Applications of Fuzzy Sets to Systems Analysis, Birkhauser Verlag, Basel 1975. L.A. Zadeh: Fuzzy Lo^c, IEEE Computer, pp.83-93, April 1988. CC. Lee: Fuzzy Logic in Control Systems: Fuzzy Logic Controller (Part I), IEEE Trans. Syst. Man Cybern., vol. SMC-20, n.2 (1990) T. Takagi and M. Sugeno: Fuzzy Identification of Systems and Its Applications to Modelling and Control, IEEE Trans. Syst. Man Cybern., vol. SMC-15, pp. 116132 (1985)

CHAPTER 2

ON THE VALIDATION OF FUZZY KNOWLEDGE BASES Didier Dubois and Henri Prade Institut de Recherche en Informatique de Toulouse Universite Paul Sabatier 118 route de Narbonne, 31062 Toulouse Cedex, France

Abstract Roughly speaking, a knowledge base is "potentially inconsistent" or incoherent if there exists a piece of input data which respects integrity constraints and which leads to inconsistency when added to the knowledge base. In the paper we use the framework of possibility theory in order to discuss this problem for fuzzy knowledge bases. More particularly we consider the case where such bases are made of parallel fuzzy rules. There exist several kinds of fuzzy rules : certainty rules, gradual rules, possibility rules. For each kind, the problem of "potential consistency" appears to be different. Only certainty and gradual rules pose serious coherence problems. In each case we caracterize what conditions parallel rules have to satisfy in order to avoid inconsistency problem with input facts. The expression of a fuzzy integrity constraint in terms of impossibility qualification is discussed. The problem of redundancy, which is also of interest for fuzzy knowledge base validation, is briefly addressed for certainty and gradual rules.

1 - Introduction Consistency is a key concept in classical logic. In fuzzy set-based logics, consistency becomes a matter of degree. Zadeh (1979) has defined the consistency of a fuzzy set A' with respect to another fuzzy set A, supposed to represent the available knowledge about a variable X (namely "X is A"), as the height of the intersection (pointwisely defined by the min operator) of A and A'. In other words, the degree of consistency refers to the existence of an interpretation (here a possible value of X) compatible with the available knowledge 31 S.G. Tzafestas and A.N. Venetsanopoulos (eds.) Fuzzy Reasoning in Information, Decision and Control Systems, 31-49. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.

32

and compatible with the statement "X is A'". The fuzzy set representing the available knowledge is interpreted as a [0,l]-valued possibility distribution which restricts, in an elastic way, the more or less possible values of a variable, or more generally of a tuple of variables. If this possibility distribution is not normalized, it means that the available knowledge is (partially) inconsistent by itself since there does not exist any fully possible interpretation (i.e. possible at the degree 1). For instance, in possibilistic logic (Dubois, Lang and Prade, 1989, 1991b), the complement to 1 of the height of the possibility distribution representing the whole knowledge base defines the degree of inconsistency of the knowledge base. Although inference under partial inconsistency can still be performed in possibilistic logic, checking the consistency of a fuzzy knowledge base remains an important issue for validating the base. The validation of fuzzy knowledge bases has received little attention until now, with the exceptions of Hall et al. (1988), Yager and Larsen (1990), Larsen and Nonfjall (1989,1991), Turksen and Wang (1992), Kinki6ia6 (1992), and Dubois and Prade (1991b). In this paper, we more particularly consider the problem of the "coherence" of a collection of fuzzy rules, where "coherence" means that there does not exist any input fact, in agreement with the integrity constraints, that together with the base of rules leads to a partially or totally inconsistent knowledge base. We thus extend to the fuzzy case ideas discussed by Nguyen et al. (1985, 1987), Ayel and Rousset (1990) among others for rulebased expert systems. Coherence conditions are given in Section 3 for different kinds of fuzzy rules which capture the ideas of graduality, certainty, and possibility of fulfilment of properties respectively. The issue of redundancy is also briefly discussed for fuzzy rules in Section 4. First, Section 2 gives the necessary background on the fuzzy set-based approach to approximate reasoning, extends the idea of coherence with respect to input data to fuzzy knowledge bases, and discusses the representation of integrity constraints which are stated in terms of impossibility qualification.

2 - Approximate Reasoning in Possibility Theory and Coherence 2.1 - Possibilistic Approach to Approximate Reasoning In the possibilistic framework, each piece of knowledge is represented by a possibility distribution K[. A possibility distribution expresses a (fuzzy) restriction on the possible values of tuples of variables. For instance, the pieces of information "X is A", "if X is B and Y is C then Z is D", where A, B, C and D denote subsets which may be fuzzy, are represented by the possibility distributions Vu, nx^*^) ~ ^ A ( " ) ' ^""^ ^ " ' ^' *• Kx Y z("'^'*) ~ (^^B(") * Mc(^)) ~^ I^DM (where * and —> are respectively conjunction and implication operators to be determined according to the intended semantics). Thus a piece of knowledge is represented by a possibility distribution on the Cartesian product of the domains of the involved variables ; see (Dubois and Prade, 1991a) for an overview and a bibliography on fuzzy set-based approximate reasoning. A knowledge base ^ made of n

33

pieces of knowledge represented by the possibility distributions Ttj, i = 1 ,n, is associated with the possibility distribution ;c% obtained as the conjunction of the Ttj's, i.e. 7C^ = mini=l^n7ti.

(1)

This is Zadeh (1979)'s combination principle, which is followed by a projection of ;r^6^, in order to compute the induced restrictions on the variables of interest. Namely if Tc^r involves the variable Xj, ..., X^, i.e. Tt^r, = 7Cxi,...,X > ^^^ '^ ^"^ ^ ^ ' ' ° compute the existing restriction bearing on a subset ?6
"k) = Proj(7t^ ; ^ ) ( u i s"Puj:Xj^X

Uk) = J^Xi

Xj

xj^l

"m)-

(2)

2.2 - Coherence and Integrity Constraints Let J^ be a base made of pieces of knowledge represented by possibility distributions defined on the Cartesian products of the domains of the involved variables. This set of variables is supposed to be partitioned into input variables X j , ..., Xjj, and non-input variables Yj, ..., Y^. The input variables are those variables about the value of which information may be received, this information being added to ?v in order to deduce new possibility distributions restricting variables of interest. The min-combination of all the pieces of knowledge in ^ leads to the definition of the global possibility distribution ' ' ^ = ''Xi

Xk,Yi

Y^

which represents the contents of fK-. Then by projecting TT^ over X j , . . . , Xj,, we obtain the restriction (if any) which is induced by n^ on the input variables ; this is called "reflecting on the input" by Yager and Larsen (1991). The knowledge base fPO will be said to be coherent with respect to input data if this projection does not induce any genuine restriction on the value of the input; namely, ^ is said to be coherent if Vui, ..., Vuk, supvj . .v^ 'txj

Xk, Yi

Y / " 1 . •••' "k- ^l- •••. v^) = 1.

(3)

Indeed, if (3) does not hold, it means (in case of finite or bounded domains that there exist input data (u^j u \ ) such that TCX, Xu. Y, YA(U^1 " \ ' ^ 1 ' •••> v.?) < 1, Vvj, ..., Vv^, i.e. that there exist input data that together with 7i<j^ make a partially inconsistent knowledge base, since the corresponding possibility distribution is no longer normalized. Note that (3) entails that, for each X,, proj(7t^,Xj) is the possibility

34

distribution uniformly equal to I on the domain of X;; but the converse is false, it is not sufficient to have the property for each projection on Xj, i = 1 ,n to insure (3). In practice, any tuple of input values is not always allowed. For instance, let us imagine that the variables Xj, ..., Xj^ pertain to the description of people, e.g. Xj = age, X2 = weight, then the data Xj < 1 year, X2 > 20 kg are not feasible. For detecting input data which would not be meaningful, integrity constraints describe the tuples of values which can be encountered and accepted. Let IC be the subset, defined on the Cartesian product of Xj, ..., X|(, of the tuples of values which are acceptable. Let us assume first that IC is an ordinary subset. Then the coherence of ^ with respect to input data which satisfy the integrity constraints is naturally expressed by modifying (3) into V (ui

u^) e IC, supvj„..v^ JtXi

X^X»l

"k- ^i, ••-, v^) = 1.

(4)

It may be natural to allow for flexible integrity constraints, which enables us to distinguish between input data which are completely possible from input data which are not completely impossible but are more exceptional, letting IC be a fuzzy set. Then we may think of extending (4) in two different ways : - by requiring that (4) holds for all tuples {\i\,.... uj^) belonging to the support of IC, i.e. Vui,...,Uk, Hic(ui,...,Uk) > 0 ^ supvj,...v^ 'tXi,...,Xk("l'--- "k-Vb--. v^) = 1. (5) In this case, the introduction in ^ of input data, even if they are almost (but not totally) impossible according to IC, will not create any inconsistency, but the fuzziness of IC is not taken into account by (5). However the fuzzy nature of IC may still be used for making clear to the user how much the input looks exceptional and has to be checked ; - by using the standard fuzzy set inclusion, namely Vul,...,u,^,^lIC(ul

Ufc) < supvj^

v^ ^Xi

XfcCui. •••. Uk' ^ i . . . - v^).

(6)

With this condition, which is weaker than (5), it is allowed that the low level of possibility of encountering some tuple of input data can be already acknowledged by JK itself provided that the level of possibility induced by X is at least equal to the one given by the integrity constraints. However, with (6) there may exist input data with a non-zero degree of membership to IC which make ^ partially inconsistent (when added to^).

35 2.3 - Impossibility Qualification for Specifying Integrity Constraints The (fuzzy) set IC can be obtained as the result of the logical conjunction (using min operation) of elementary integrity constraints. Usually elementary integrity constraints are Slated in terms of impossibility rather than in terms of what is possible, like for instance, "X is A is forbidden, is impossible", or more generally "X is A is almost impossible" (where X is a variable or a vector of variables). The integrity constraint C "X is A is impossible to the degree a" can be translated as a constraint on the induced possibility distribution Jt;;^ describing the allowed values of X (the conjunction of such possibility distributions leading then to the definition of m^). Let us first consider the case where A is an ordinary set n(A) = supyg \ Jto(") S l - o t < = > V u 6 A, 7t(^,(u) ^ 1 - a i.e. (when a = 0 there is no constraint on ir^ and when a = 1, all the values in A are excluded as expected) When A is a fuzzy set, two interpretations may be considered : i) we understand "X is A is almost impossible" as "X is A is almost certain" (where A denotes the (fuzzy) complement of A). Then, we can apply the results about certainty qualification (Dubois and Prade, 1991a), which yield jr(;(u) = max(n;^(u), 1 - a) = max(l - \i.xi^), 1 - a) = 1 - min()U;s^(u), a).

(7)

The result is pictured on Figure 1. ii) starting with the usual definition of the possibility of a fuzzy event leads to write n(A) = supu min(HA(u), 7t'c,(u)) < 1 - a i.e. Vu, min(^y^(u), 7i'(^(u)) < 1 - a i.e. Vu, 7:'c(u) <

[HA(U)

-»(I - «)] with a - ^ b = ( ' ' f ^ - ' ' lb if a > b.

Allowing the greatest possibility degrees compatible with this constraint yields the possibility distribution Jt't; pictured by a dotted line in Figure 1, i.e.

't'c=I^A^(l-")•

(8)

36

When a = 1, n'l^ is nothing but the complement of the support of A (the set of elements with non-zero membership grades)

1-a * •

X

ijtc

Figure 1 Figure 1 exhibits the difference between n^= I - min((iy^, a) andrt't^= n^^ —> (1 - a). In the translation of "X is A is almost impossible" by n'l^, the gradual nature of A is lost (even if n't;^ does not correspond to the membership function of an ordinary subset when 0 < a < 1) ; Tt'i; corresponds to a rather drastic solution which relates the (ordinary) subset concerned by the impossibility to the level of impossibility a, via n^. We may thus prefer, to K'I^, the first interpretation n^, which reflects the gradual nature of A. 3 - Validation of a Base of Fuzzy Rules 3.1 - Different Kinds of Fuzzy Rules In Dubois and Prade (199la) a typology of fuzzy "if... then..." rules of the form "if X is A then Y is B", where A and/or B are fuzzy sets has been proposed. A distinction is made between - gradual rules whose intended meaning is of the type "the more X is A, the more Y is B" and which are represented by a possibility distribution Kyix obeying the inequality Vu, Vv, min()iA(u). tYlx("'V)) ^ M^B(V)

37

whose greatest solution is expressed by means of so-called GOdel implication (this possibility distribution allocates the greatest possible possibility degrees and thus does not reduce the range of possibilities arbitrarily)

.v«(v.«)..A<.)-.B(v).!;;'(i:fr^B':>

m

- certainty rules whose intended meaning is of the type "the more X is A, the more certain Y is B" and which are represented by a possibility distribution JCyiX 'tYIX(v.u) = max(^iB(v), 1 -

(10)

RA("))

which indeed expresses that "Y is B is |iA(u)-certain" (according to the result already applied in (7)); - possibility rules whose intended meaning is of the type "the more X is A, the more possible Y is B" and which are represented by a possibility distribution Ttyix obeying the inequality rtYix(v,u) > min(^A("). M-B(V)) (11) which indeed guarantees, for any input UQ, that the possibility that Y = v for all the values V such that \i^(y) = 1 is at least equal to HA("O)We now study the coherence of a knowledge base ^ made of a collection of parallel rules of the same kind, "if X is Aj then Y is Bj", i = 1 ,n in the case of certainty rules, and in the case of gradual rules. There is no coherence problem with possibility rules (which is intuitively natural for rules whose conclusions express only things which are possible/ feasible). Indeed possibility rules have to be combined disjunctively due to (11) (see Dubois and Prade, 1991a), and then if the Bj's are normalized supv 7CYix(v.u) >• 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 <=* cons(Bj,B2) = 1 or cons(Aj,A2) = 0-

(14)

The necessary and sufficient condition of coherence (14) for two rules can be equivalently written core(B j) n core(B2) ^ 0 or support(A j) n support(A2) = 0 (14A) with supporl(Aj) = (u, 1x^(1) ^ ^) ^nd core(Bj) = (v, |1B.(V) = 1}. It expresses that the condition parts of the rules have to be completely disjoint or the conclusion parts of the rules have to be fully consistent. It can also be expressed as : whenever the supports of A j and A2 intersect, the cores of B j and B2 should intersect too. A simple example of certainty rules are rules expressed by possibilistic formulas (see, e.g. Dubois, Lang and Prade, 1989) of the form "if p then q with certainty a" where p and q are classical propositions. Such a certainty rule is represented by the possibility dismbulion Tt on the set of interpretations 9, V 1 6 3- 't(l) = 1 if I is a model of —ip v q = 1 - a if I is a model of p A -iq = max(HM(-,pvq)(l). 1 - a) = max(l - HM(p)(I), |iM(q)(^)' 1 - «) where 1J.M(') denotes the {0,1) -characteristic function of the set of models of a proposition. Thus a possibilistic certainty rule is a particular case of a fuzzy certainty rule, identifying l^M(p) ^ ' * I^A 3"d fn3x(|J.f^^(q\, 1 - a) with jig. Then for two rules "if pj then q; with

39 certainty a;", the condition (14) only holds as soon as M(pj) n M(p2) ^^ 0 entails M(q]) n M(q2) * 0. For instance the two rules "if p then q with certainty a" and "if r then —iq with certainty 3" cannot be coherent together as soon as p and r are not mutually exclusive since then cons(Bi,B2) = niax(l - a, 1 - p) < 1 (except if min(a,3) = 0). Let us now consider the case of a knowledge base % made of n parallel rules. Then proj(^ ; X)(u) = supv min;^] „ max(l - |iA,("). V^^f"^)) = supv maxj^l[i,nI,0
HAJ^")-

VL^^M),

HB|^(V))

|UB|^(V))

H A - W ' •'^"PV l^B-W)'

supv min(^B (v), IIB^CV))), •.,

^uPy "I'lj^ii t^BjW)- ^uPv "i'"i=l,n ^iBiM^

= max[1 - ^Aiu...uAn("). "^a^j min(l - n y . A.(U), hgt(Bj)), maxj k min(l - ^Uj^j j ^ ^ Ai("). hgt(Bj n B^)) max, min(l - 11AJ(U), hgt(nj;ii Bj)), hgt(Bi n...(^ B^)] (15) with hgt(F) = supv JIF(V), and obviously hgt(F n G) = cons(F,G). Let P be the partition induced on the domain of X by the supports of the Aj's, i = 1 ,n. Each subset in the partition is of the form (Hie j support(Ai)) o (Rig i support(Aj)) where I c [l.nl. Thus if we consider an element u belonging to such a subset, we have u e rijg 1 support(Aj) => \i\j

^U\x) = 0 => min(l - txy.^ JAJW. hgt(nie I ^i)) = hgt(nie I Bj).

(16)

Moreover, noticing that 1 - IIAUFC") = I => 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 <^ B3)), min(l - HAJ^")- hgi(Bi n B3)), min(l - ^A3(")' hgt(Bi n B2)), hgt(Bi n B2 n B3)). Then it can be checlced that % is coherent if and only if hgt(Bi 0 8 2 0 8 3 ) = 1 or hgt(Bi o B2 o 83) < 1, hgt(Bi o B2) = hgt(B2 o B3) = hgt(Bi o B3) = 1, A] o A2 o A3 = 0 or V i #j 9t k, hgt(Bi o Bj) = hgt(Bj o Bj^) = 1, hgt(Bi o Bj^) < 1, Aj o Aj^ = 0 or V i ^ j ;* k, hgt(Bi o Bj) = 1, hgt(Bj o Bj^) < 1, hgt(Bi o B|^) < 1, hgt(Bi(^) = 1, (A; u Aj) o Ajj = 0 or V i ^ j ^ k, hgt(Bi o Bj) = 1, hgt(Bj o Bj^^) < 1, hgt(Bi o B^.) < 1, hgt(Bij) < 1, Ak = 0 or hgt(Bi o B2) < 1, hgt(B2 o B3) < 1, hgt(Bi o B3) < 1, hgt(Bi) = hgt(B2) = hgt(B3) = 1, (Aj u A2) o (A2 u A3) o (A) u A3) = 0 or V i 5* j ;i k, hgt(Bi) = 1, hgt(Bj) = I, hgt(Bi o Bj) < 1. hgt(Bk) < 1, (Aj o Aj) u Aj^ = 0 or V i ^ j * k, hgt(Bi) = 1, hgt(Bj) < 1, hgt(Bk) < 1, Aj u Aj^ = 0 where fuzzy set intersection and union are defined using min and max respectively. As we can see the condition is already much more complicated than in the case n = 2, even if it can be made simpler with the natural hypothesis that the Bj's are normalized, i.e. Vi,

41 hgt(Bj) = 1. Namely, in the nonnalized case, the above condition expresses that if the cores of 81,62,83 have not a common part (hgt(Bi 0 8 2 1 ^ 83) * 1), the three parallel certainty rules are coherent if and only if Aj n A2 n A3 = 0 , and as soon as hgt(8j n B^^) < 1, Ajpi Aj. = 0 holds, Vj, Vk, or if we prefer, ni=i,3 support(A;) ^0=» ni=i,3 core(Bi) * 0 and support(Aj) n support(A|j) * 0 => 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<7r.

(21)

43

Indeed if (21) holds, we have min(minj_2 ^ TIJ, n) = TC^ and thus n brings nothing new to % . Moreover if % is coherent in the sense of (3), the addition to ^ of a redundant piece of knowledge leaves the knowledge base coherent obviously. Let us examine the meaning of (21) in case of knowledge bases made of parallel certainty, or of parallel gradual rules.

4.1. Certainty Rules First, consider the case of the redundancy of one certainty rule "the more X is A, the more certain Y is B", with respect to the other certainty rule "the more X is Aj, the more certain Y is Bj". (21) writes Vu, Vv, max(l - ^Al(u). PiBi(v)) ^ max(l - HA(U). UBC^))

(22)

i.e. Vu, Vv, 1 - HAI(U) ^ •"ax(l - HA.(U). IJ-B(V)) and liBj(v) < max(l - \i/^(u), J I B M ) and then, provided that A is normalized, we get ^ 3 - < p-g and provided that 3v, |a.B(v) = 0, we get lO-A - HAI • ^-^^ redundancy is, as expected, equivalent to A c Aj and B | c B. If infy Hg (v) = X > 0, the redundancy conditions write Bj c B and (1^1 - m'n(UA'' " ^)' or equivalently, n ^ ^ (1 - ' ' • ) - * M^Ai ^^^ a - > b = l i f a < b and a ^ b = b otherwise. Let us now consider the redundancy of a certainty rule with respect to a set of two other certainty rules, i.e. (21) writes Vu, Vv, mini=i_2 max(l - HAJ(U), |iBj(v)) ^ max(l - n^Cu), PB(V))-

(23)

Applying distributivity, it is equivalent to Vu, Vv

1 - HAIUA2('^) - '"^''(^ - M " ) ' ^LB(V))

and and

min(l - HAJC"). ^iB2(v)) ^ ^ " ( 1 - ^^A(u)• liB(v)) min( 1 - HA2(")' ^B J (V)) < max(l - PA(U). HBC^))

a^

I^BinB2W - ""^^(^ - M " ) ' ^lB^))•

Then, if A is normalized, redundancy entails Bj n B2 c B, and if 3v, |IB(V) = 0, it also entails Aj u A2 3 A. Hence if Bj n B2 ^ B, or if A| u A2 ^ A (if infy HgW - 0)'

44

we are certain that the rule is not redundant. Besides, if the rule is already redundant with respect to one of the rules in the base, the rule is obviously redundant with respect to the base. Moreover the two rules "the more X is A;, the more certain Y is Bj", i=l,2, entail, on the one hand, that "the more X is Aj and A2, the more certain Y is B j and B2", and on the other hand, that "the more X is Aj or A2, the more certain Y is Bj or B2", since it can be easily proved using distributivity and obvious boundings that •"'"1=1,2 '"ax(l - liAi(u).tiBi(v)) ^ min[max(l - |iAinA2(")' l^BinB2(^))' rnaxd -HAluA2•^lB,uB2W)]• Thus, if the rule "the more X is A, the more certain Y is B" is redundant with respect to one of the two above rules which are consequences of the knowledge base, the rule will be redundant with respect to the base (by transitivity of <). Thus the redundancy of a certainty rule with respect to one of the two certainty rules in a knowledge base, or with respect to the rules generated from this base (either by intersection of the conditions and of the conclusions, or by their union), entails the redundancy of the certainty rule with respect to the knowledge base in the sense of (23). However the converse is generally wrong. The situations is better for rules with non-fuzzy condition pans. Indeed if A J and A2 are crisp subsets, the content of the knowledge base is represented by liBinB2(v)''""e , , ,UBi(v) ifuG JtX(u.v)= \ mB2(v) if u e U if u 6

Ai Ai -7Ai Ai

0A2 nA2 ^ n A2 n A2 .

Assume moreover that A is normalized and M.B(V) = 0 for some v. Redundancy writes Vu, Vv, 7t:^(u,v) < max(l - HA(U)' l^B^^)) ^"'^ ^^^^ '' imp'ies A c A] u A2. Let us assume that A is normalized. Thus, i) if A c A] n A2 redundancy implies B | n B2 c B, ii) if A c Aj and A 5^ A] n A2, i = 1 or 2, it implies Bj c B, and iii) if A c Aj u A2 but A ^ A^, i=l,2, then Bj u B2 £ B provided that 3 u' e A] n A2, |iA("') = ^ ^^^ ^ u" e A] n A2, = 1- In this case, the rule is redundant with respect to one of the two rules "if X is Aj, then Y is Bj", or with respect to one of the two other rules obtained from the two previous rules via intersection or union operations. But it may happen that either u' or u" does not exist, i.e., 0 < supyg ATOAT I^A(") "^ I^A(" ) *- ^ ^'^^ instance ; then we only obtain Vv, [13 j(v) < H B W ^nd [i^^'^'^) < max(l - |aA(u"), HB(V)). '•£• liBiuB2 ^ ^ " 0 |J.A(U")

|JA(U "). HB)- ^ " ' ^^^^ '^'' ^^^y special situation. When Aj or A2 are fuzzy, the redundancy with respect to the base is not equivalent to the redundancy with respect to one the four rules mentioned above (those of the base or

45

those generated via intersection or union). Indeed, let us consider the case where B j and B2 are ordinary subsets. For a fact X = UQ, the knowledge base yields Tty = min(max(n3 1 -ttAi("o))' "i^xCja-g-, 1 - HJI^JUQ)). Suppose that for instance ^ A I ( " O ) -* IIA^CUQ) > 0. Then clearly ny < max (jig , ^ 3 ^ , 1 - ji^^(uo)) in general, the upper bound being the result obtained by means of the rule, "the more X is Aj n A2, the more certain Y is Bj n B2". Thus, as soon as the rules have fuzzy conditions, there does not always exist a rule which is redundant with respect to a rule base and which can produce the same conclusion as the rule base for a given input.

4.2. Gradual Rules In order to make a first step to the redundancy checking of gradual rules, let us examine conditions under which a rule of the form "the more X is A, the more Y is B" is redundant with respect to another gradual rule. A necessary and sufficient condition is given by applying (21), namely Vu, Vv, ^Aj(u) -^ ^Bi(v) 5 liA(u) -^ liB(v)-

(24)

It is easy to check that a—»b>c-^d<=> ( a < b ) o r ( a > b and b > c ^ d) for Godcl implication. Hence (24) is equivalent to Vu.v, ifnA,(u) 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.

This page intentionally blank

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<:h bending point with an element in universe of discourse and corresponding grade of membership. A fuzzy set with triangular membership function can be described a^ {0/0-0/2.5-1/4-0/5.5-0/10}

(3)

where its universe of discourse is [0,10] interval. Linear interpolation is normally supposed to assure that grade of membership can be obtained for every element in the universe of discourse. This method has the same problem with description in functions. Results of operations can not necessarily be represented in pointwise description. Vector Representation(Fig. l-(c)) More general and most widely used approach is to represent a fuzzy set in a vector of grades. If the universe of discourse is defined and discretized by 0.5 step, the previous fuzzy set is expressed with a vector [0,0,0,0,0,0,0.33,0.67,1,0.67,0.33,0,0,0,0,0,0,0,0,0,0]. (4) In this method, operations on fuzzy sets are easier than previous two methods. Fuzzy set operations such as union or intersection can be executed simply as element by element operations between two vectors, and their results are in the same form. This kind of fuzzy set representation is often used in expert shells for fuzzy control for other fuzzy inference based systems. For fuzzy control systems, universes of discourse are usually fixed range in numeric scale, and operations needed for fuzzy inference are variations of union and intersection. Moreover, control shells usually need fast execution of inference. Fuzzy set operation in vector representation is suitable for this purpose. Paired Representation(Fig. l ) - ( d ) ) Most general type of fuzzy set representation is this paired representation, vector representation does good if universes of discourse are fixed ranges in numeric scale, but fuzzy sets in general are not restricted by definition. For example, a fuzzy set of beautiful women can be {0.8/'Ann', 0.6/'Beth', 0.9/'Cathy',-• •}, though its universe of discourse cannot be determined. Another case happens when the universe of discourse consists of

54 fuzzy sets. This kind of fuzzy set is called 'level-2' fuzzy set and is difficult to be described in ottier ways. In paired representation, fuzzy sets are described like, yl = / z , / u i + / / 2 / W 2 + ••• + /'r,/«n

(5)

or more generally, n

A = Ylnilui.

(6)

This method is flexible enough to permit almost all kinds of fuzzy sets. Q Level Sets Representation(Fig. l - ( e ) ) crisp set defined as

An a level set of a fuzzy set A\s a.

A^ = {u\iiA{u)>a,ueU)

(0
(7)

And a fuzzy set can be represented with a family of these a level sets as A

/

ccA,

(8)

^ €>€(0,ll (0,1]

and if the fuzzy set is convex, each a level set becomes an interval in U. So a fuzzy set are described with a set of numeric intervals. With this representation, fuzzy number operations can b e handled in a simple way. For example, a two operand operation * between two fuzzy sets A and B can be executed as A*B=I {A-kB)c= I A^*Ba (9) •^ae(o,il -'oe(o,i] and the right hand side can be computed as operation on intervals.

3.

A Brief Survey on Fuzzy Logic Based Softwares

Here a brief survey is made about software systems based on fuzzy set theory and fuzzy logic. Many systems have been proposed and they can be clcissified in three categories. 3.1

S o f t w a r e for M a n a g e m e n t of U n c e r t a i n t y

T h e first category is software systems for management of uncertainty. One of the earliest works is PRUF[1] which was proposed by Zadeh. T h e main aim of P R U F was to represent the uncertainty in natural language in a formal manner. P R U F has its hash on possibility distribution rather than t r u t h value. Uncertainty or vagueness in natural language is represented in terms of possibility distributions and fuzzy propositions. T h e main interest in this system was the representation of natural language on computer systems, using fuzzy logic as a tool for managing uncertainty.

55

(b) Pointwise

(a) In Function

<1 • • •

T

•

• I• •

• ; : ;•

• ; ; I ! !•

•

• ! ! ! : ! : • •

'1'l»»»< U (d) Paired Representation

(c) Vector Representation

•

•

(e) a level sets Figure 1: Several Types of Fuzzy Set Representation

56

L.P.L. language[2, 3] is also a system for managing uncertainty in programming language. It has the ability of treating fuzzy predicates and control structures with fuzzy notion. These control structures help to build linguistic and fuzzy algorithmic procedures. Another approach toward management of uncertainty wa.s to introduce fuzziness in Prolog language. As Prolog is based on the first order predicate calculus, so fuzzified Prologs are based on fuzzy logic. There are several levels of fuzzification. Most fundamental level is to allow fuzzy value(in [0,1] interval) for truth values of clauses and predicates. This type can be extended to more flexible systems that allow fuzzy values for all Several systems are such a.s FS-Prolog[5] and Fprolog[4]. There are also some expert systems that have the notion of fuzziness in certainty factor or in degree of confidence. FL0PS[6, 7, 8] is one of these systems. FLOPS allows 'fuzzy' values for confidence of data and relation between left hand side and right hand side of rules. This approach can be compared to CF(certainty factor) in expert systems like MYCIN. Other parts of expert systems, such as condition part of rules or data itself, can also have fuzziness. For example, a rule "If X is rather tall then • • •" or a fact "Peter is short" are admitted in some expert systems. System Z-II[9] can handle this kind of uncertainty using fuzzy set theory and certainty factor. 3.2

Fuzzy Control Shells

Many control shells have been proposed to support the development and execution of fuzzy inference used for fuzzy control systems. Most of them consist of graphical user interface for editing fuzzy rule base and membership functions and sometimes a simulator or knowledge acquisition tools, working as development tools. Some systems are equipped with dedicated hardware for high-speed processing of fuzzy inference, as fuzzy control needs high speed execution for some applications. Fuzzy-C Compiler system[10] is one of the most refined systems of this kind. Bcisically Fuzzy-C Compiler is a translator from Fuzzy Programming Language(FPL) to C language, which is oriented toward fuzzy control systems and there are tools for building fuzzy control systems. The most distinct feature of this system is that hardware solution is also provided. There are specially designed chip for fuzzy inference. Unlike other fuzzy logic chips, the chip(named FC-110) works with a machine language which provides not only fuzzy inference but also other fuzzy logic related instruction. 3.3

Fuzzy Set Representation

The last category is software systems designed for fuzzy set processing in general. They are usually based on some standard programming language, taking fuzzy set processing into account. These systems are motivated by the needs for flexibly manipulating fuzzy theory based systems with software. One of the earliest works Wcis FSTDS (Fuzzy Set Theoretic Data Structure)

57

system[ll]. FSTDS is an extension to Fortran and can represent and manipulate fuzzy sets in general. It has a pointer-connected data structure and the data structure is using the 'paired representation'. By providing 'tuple area' for elements and grades, this system realized the representation of complicated fuzzy sets like L-fuzzy sets or type-n, level-m fuzzy sets. A Lisp based "Fuzzy Set Manipulation System in Lisp" [12] was designed later by the same research group. This system is provided as an extension to Lisp and represent fuzzy sets in paired style. By using tuple for representing grades of L-fuzzy sets and elements of relations, general representation of fuzzy sets are achieved. This system also provides a simple way to describe fuzzy sets in ordinally Lisp programs. Another Lisp based system FLISP[13, 14] provides similar data structure for fuzzy sets.

4.

Object-Orientation in Fuzzy Software

Many software systems have been designed for fuzzy information processing, as described in the previous section. But these systems have limited capabilities in the sense they are internally representing fuzzy sets in a single type of data structure. Fuzzy sets can be represented in many ways and each representation ha^ different characteristics in its abilities in representation and costs for memory and computation. So it is very difficult to represent every kinds of fuzzy sets in one specified method, and the integration of various fuzzy set representations are required. In addition, operations on fuzzy sets heis wide variety. Representation of fuzzy sets requires complicated manipulation over data-structures. Because a datastructures and procedures for them are described separately in many procedural languages, the corresponding procedures must be described separately for each data-structure. It is therefore difficult to ensure the consistency of descriptions of a data structure and accompanying procedures. Object-oriented paradigm, or object-orientation, is a framework in software engineering. The essence of object-oriented paradigm is to represent elements in systems as 'objects' that know how to act themselves, and to execute a program by sending messages to objects. By representing fuzzy sets as objects, the demand for integrating fuzzy set manipulation and numerical or symbolic processing is considered to be solved; thus fuzzy set processing is integrated with other information processing techniques. On such idea, a software system named FOPS was developed. This system is based on the Smalltalk[15] language and intended to be a prototype of an objectoriented software for fuzzy information processing. This system is to demonstrate the effectivities of object-oriented approaches to fuzzy set manipulation, serving as an experimental tool for fuzzy information processing.

58

5.

FOPS: a Fuzzy Set Manipulation System

F0PS[16] is a system for manipulating fuzzy sets and it provides a way to represent and manipulate fuzzy sets on programming language basis. FOPS is constructed on Smalltalk system which is a representative object-oriented language. T h e Smalltalk system is said to be an environment as well as a language, with its a b u n d a n t class libraries and sophisticated development environment. By adding the functions for fuzzy set manipulation on it, an environment suitable for experimental implementations of information processing with fuzziness is realized. F O P S is designed to satisfy following demands; • Easy description of fuzzy sets, • Versatile representation of fuzzy sets, • Representation of fuzzy relations, • Choice of internal representation and their integration. To represent fuzzy sets in systems satisfying these demands, some classes are defined which deals with creation and manipulation of objects t h a t represent fuzzy sets. Also, some tools are built, like inspectors (tools for looking into objects) specialized for fuzzy sets. Changes in original system are small, except for the change in parser for fuzzy set description and system defined methods of max: and min: for class Number, preserving the benefits of Smalltalk system. 5.1

R e p r e s e n t a t i o n of F u z z y S e t s

I n t e r n a l R e p r e s e n t a t i o n s Two basic classes. ArrayedFuzzySet and PairedFuzzySet, and other derivations are provided to represent fuzzy sets in F O P S . ArrayedFuzzySet(AFS) ( F i g . 2 - ( a ) ) This class is for fuzzy sets with arrayed representation. This class realizes a method of fuzzy set representation that keeps grades of membership in an array, which is in general a n-dimensional array in n-ary fuzzy relation. Its universes of discourse are represented with a sequence of indexable collections (SequenceableCollection). this representation is effective in such cases that universes of discourse can be specified or greater portion of elements have non-zero grade. It is also effective when universes of discourse are intervals of number, with the compact representation of intervals. Another class specialized for linguistic truth values are defined as LinguisticTruthValue(LT\/) as one of its subclcisses. PairedFuzzySet(PFS) ( F i g . 2 - ( b ) ) This class is for fuzzy sets in paired representation. In this class, fuzzy sets are represented with a linear list of pairs, each pair hold an element in universe of discourse and corresponding grade. This representation is effective when universes of discourses can not be specified or small

59 Array of Membership Values (gradearray)

Universe of Discourse (universes)

0

0.3 0.7 0.5

0

1

0

0.3

0.7

0.5

0

1

{0.3/Ca'a.&5), 0.7/('a'&&7), 0.5/Cb-&&3), 1/Cb'&&7) from: #('a' 'b') to: (3 to: 7 by: 2)} (a) Arrayed Fuzzy Set

0.3 0.7 'a'&&5 r - ^ ' a ' S S 7 1

0.5 'b'fi£3

'b'Sfi7

2

a'l 5 {0.3/('a'&&5), 0.7/Ca'&&7), 0.5/Cb'&&3), 1/Cb'&&7)} (b) Paired Fuzzy Set Figure 2: Fuzzy Set Representation in F O P S ratio of elements have non-zero grade. In each representation, elements in universe of discourse and grades are represented by objects as well as the fuzzy sets. In plain fuzzy sets (grades and elements are not fuzzy), elements and grades are numbers like Integer or Float, but they can also be fuzzy sets, yielding type-2 fuzzy sets and level-2 fuzzy sets. More general fuzzy sets are produced with recursive application of this step. D e s c r i p t i o n of F u z z y S e t s Fuzzy sets are described with following form of expression in F O P S . Segments of text enclosed by ' { ' and ' } ' are interpreted as 'fuzzy set expression', in which binary methods ' / ' and ',' are interpreted as separator between grade and element and summation operator in fuzzy set, respectively, syntax for fuzzy set description is shown in Figure 3.

60 primary

variable name Vw.

literal block

s<(>.4 ^•({V-^ fuzzy set expression

K>>-"

expression fuzzy set expression

"*Vi)—

empt'i \

•

grade — ^ v / y ~ ^ elements

-<

O uraverses ^-^

'

ON:

- ^

y-Hltv:|

> from: — •

7

•

step

universes — • t o : - •

universes

y

fr universe «£ elements

^

element

W^-^

a

element fi£

-H)

grade, element, universe, step: primary

Figure 3: Syntax for fuzzy set description

61

An expression is internally expanded into a sequence of message expression, the result falls into an object whose class is either of AFS, PFS or LTV. Expressions without description about universes of discourse yield PFS, expression with on;, ON: or from:to: yield an AFS (with fromito:, the result is a fuzzy relation). Expression with Itv: or LTV: yield an LTV. With message in capital letters (ON: and LTV:), linear interpolation are done in object creation. 5.2

O p e r a t i o n s on Fuzzy Sets

Operations on fuzzy sets are executed by sending messages to fuzzy sets. They are implemented as 'methods' of classes. It will be confusing if each of the various kinds of classes has different way of manipulation, or if they are not compatible in operations. Conversion between classes are supported to solve this problem. PFS has no information about its universe of discourse and its structure is more flexible than AFS. So it can not be converted into AFS without information about its universe of discourse. Message like on: or fronn:to: is used to give information about universe of discourse. AFS has more information than PFS and can be converted into PFS without any additional information. This conversion can be explicitly executed with a method asPairedFuzzySet. PFS is also used as a intermediate form for input/output of AFS or LTV. The notion of 'generality' in Smalltalk is used to make such conversion automatically. This indicates how universal the class is, an instance of a class of smaller generality can be converted into one of larger generality. When a operation is executed between two fuzzy sets of different kinds of representation, messages for conversion(coercion) is issued to fuzzy set with smaller generality. After the two fuzzy sets get to be of the same type, appropriate operation is executed. These steps are the same with the way Smalltalk system handles the subclasses of Number, that is one of the reasons that fuzzy sets defined here can be treated in the same way with system defined number objects. Some samples of coercion are shown below. 3+{0/l, 1/2, 0/3 ON: (1 to: 3 by: 0.5)}. " {(l/2)/4.5, 1/5.0, (l/2)/5.5 on: (4.0 to: 6.0 by: 0.5)}" Integer and ArrayedFuzzySet." 2 * {0.3/0.8, 1/1.3, 0.6/1.6}. " {0.3/1.6, 1/2.6, 0.6/3.2}" Integer and PairedFuzzySet." {0.4/0.3,1/0.4,0.5/0.5} union: {0/0.5,1/0.8 LTV: 0.1}. " {0.4/0.3, 1/0.4, 0.5/0.5, (l/3)/0.6, (2/3)/0.7, 1/0.8, 1/0.9, 1/1.0}" PairedFuzzySet and LinguisticTruthValue"

62

5.3

Linguistic Truth Value

Linguistic truth values are widely used in fuzzy theoretic information processing. It is a kind of fuzzy sets, so it can be represented by either of AFS or PFS. But linguistic truth value has some characteristics in particular, namely, their universe of discourse is confined to [0,1] interval and some special operations are defined. Then a class LTV is defined as a subclciss of AFS. They have fixed universe of discourse([0,1]) and methods are re-defined in LTV to keep their universes of discourse of the result of operations. A sample is given in which type-2 fuzzy sets are described with linguistic truth values. small := {1/0, 0/0.4 LTV: 0.1}. middle := {0/0.1. 1/0.5, 0/0.9 LTV: 0.1}. large := {0/0.6, 1/1 LTV: 0.1}. fsetl : = {large/'a', large/'b' on: # ( ' a ' 'b')}. fset2 : = {middle/'a', small/'b' on: # ( ' a ' 'b')}. "fsetl intersection: fset2.

•• {{(l/4)/0.2, (l/2)/0.3, (3/4)/0.4, 1/0.5, (3/4)/0.6, (l/2)/0.7, (l/4)/0.8 Itv: 0.1}/'a', {1/0.0, (3/4)/0.1, (l/2)/0.2, (l/4)/0.3 Itv: 0.1}/'b' on: # ( ' a ' 'b' )}" 5.4

Fuzzy Relations

A fuzzy relation is a fuzzy set whose universe of discourse consists of a Cartesian product of collections. A notion of Sequence is used to describe the dimensions in universes of discourse. Fuzzy relations are represented by making a Sequence be the universes of discourse (and elements in them). Logical operations are valid for fuzzy relations, and some relational operations such as composition, projection, cylindrical extension are supported. Samples of fuzzy relational operations are as follows, fsetl := {0.1/'a',0.3/'b',0.5/'c'}. frell := {0.2/('a'«i&l),0.5/('a'A!Si3), 0.9/('b'.!^&l),0.3/('c'&&2)}. "fsetl compose: frell

" {0.3/1, 0.3/2, 0.1/3}" frel2 := {0.4/(li^&l),0.6/(l&&2),0.7/(l
" {0.9/(l&&l),l/(l&&2),0.7/(l&&3),l/(I&&4), 0.9/(2&&l), l/(2&&2),0.7/(2&&3),l/(2&&4), 0.9/(3&&l),l/(3&&2), 0.7/(3&&3),l/(3&&4) on: (1 to: 3) kk (1 to: 4)}"

63

Figure 4: Tools in FOPS 5.5

Tools for Interface

Many software development tools, such as browsers and debuggers, are provided in Smalltalk system. Tliey can also be used on FOPS and provide a. good programming environment. Furthermore, some tools specialized for fuzzy sets are provided like inspectors(Figure 4). An inspector is a tool for seeing and changing the internal states of objects. Two inspectors are implemented reflecting the structures of AFS and PFS as fuzzy sets. Fuzzy set objects have printOn: method which gives character base description of the object. The printOn: method for fuzzy sets are implemented to give strings in the form of 'fuzzy set expression', which can be evaluated by the Smalltallc Compiler. AFS has a primitive but useful methods for graphical interface. Fuzzy sets can be graphically presented on display. A class FuzzySetMonitor is prepared to display

64

fuzzy sets in windows (views). By evaluating an expression FuzzySetMonitor open: H^NameOfMonitor a new view is opened. And by evaluating NameOfMonitor monitor: fuzzySet the monitored fuzzy set is displayed in the view. Multiple monitors can be opened and can be referred to by different names. 5.6

Extensions t o t h e S y s t e m

Defining O p e r a t o r s on Fuzzy Sets Many kinds of operations are used on fuzzy sets. Just for intersection of fuzzy sets, infinite number of operators are proposed. As it is impossible to support all the operators, implementation is limited to distinguished operators, leaving other operators open to user's definition. The definitions of new operations can be easily done with the use of a method performBlockToGrade:with:. For example, to add a method for bounded difference to AFS, a new method, say bDif: can be defined cis follows. ArrayedFuzzySet methodsFor: 'logical operations' bDif: fuzzySet "self performBlockToGrade: [:a :b I (a — b) abs] with: fuzzySet

O t h e r T y p e s of Fuzzy Set As stated before, the two kinds of fuzzy set representation are ready for use in the system, but other ways of representation can be easily added on to the system. Newly created cla^ss should suffice following conditions; 1. Is one of subclasses of ArithmeticValue, 2. Operations are executable between two objects in that class, 3. Have coerce: method to convert into fuzzy set representations with higher generality, 4. Objects with lower generality can be converted into instances of the class. For example, to define a fuzzy set whose membership function is given by a function, it can be represented as a object that consist of universes of discourses and a block for the function. Some special kinds of fuzzy sets can be represented by assigning some specific class of objects to grades, because operations on grade values are interpreted by each objects representing grades. For example, representation and operations of L-fuzzy sets can be accomplished by defining a class for partially ordered objects (like tuples) and assigning them to grade values of fuzzy sets. Also, assigning interval values to grades makes the fuzzy sets can be applicable to fuzzy interval logic or inverse problems of fuzzy relations. In summary, FOPS has following features:

65

1. Easy and flexible description of versatile fuzzy sets 2. Integration of multiple internal representations 3. User-friendly interface FOPS is now working on ParcPlace Smalltalk systems (Version 2.5 and Release 4) on Sun workstations.

6.

Comparison of Representation Methods

A simple performance test was made on FOPS system to evaluate the real difference in the amount of computation between representation methods. Memory capacity and computation cost depend on the ways of representation. But it is not so easy to evaluate the difference of computation cost, because it also heavily depends on implementation. In operations on fuzzy sets, it sometimes takes much computation to get or store back a grade of membership for a given element. Then the load of computation does not depend only to the number of operations between grades of membership. In FOPS, there are two basic types of fuzzy set representation, namely arrayed representation and paired representation. Actual running times were measured on a real implementation, which will show the general characteristics of the representations. Three operations were picked up as illustrations of three kinds of operations. (a) fsetl intersection: fset2 (b) fsetl min: fset2 (c) fsetl compose: frell where, fsetl and fset2 are fuzzy sets like, AFS; {0.5/1, 0.5/2, 0.5/3, 0.5/4 on: (1 to: 4)} PFS: {0.5/1, 0.5/2, 0.5/3, 0.5/4} for 'number of element' — 4. Fuzzy relation frell is like AFS: {O.S/{liiSil). 0.5/(l&&2), 0.5/(2&&l), 0.5/(2&&2) from: (1 to: 2) to: (1 to: 2)} PFS: {0.5/(l&&l), 0.5/(l&&2), 0.5/(2&&l), 0.5/(2&&2)} for 'number of element' = 2. Intersection operation was taken an illustration of two operand fuzzy set operations. The result is shown in Figure 5-(a), which shows that AFS is about 4 times faster than PFS. This is because there is no need to make calculation for elements in universe of discourse. Figure 5-(b) is the result of min: operation, that is the illustration of operation based on extension principle. The difference is not so evident but PFS is a little faster. This is because grade operation is needed for all combinations of elements in such operations, regardless of the representation method.

66 - Time (msec) -

>

10.00-!

; yii'"'

; 5^ 2

4

8

16

32

64

126 2S6

Nuntier of elBments '

© 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).

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[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<**),

and S.Terzakis(**)

(*)

Department of Electrical Engineering University of Toronto, Toronto, Ontario, Canada M1551A4 (**) Intelligent Robotics and Control Unit National Technical University of Athens Zografou Campus, Zografou 15773, Athens, Greece 1. INTRODUCTION This chapter deals with fuzzy dynamic systems paying attention to three important issues, namely analysis, control and identification. Analysis is concerned with the study of inputoutput descriptions which are based on fuzzy relational equations. One of the models that played a central role in the analysis of classical (crisp) dynamic systems is the state-space model. This model can be extended to cover the case of fuzzy dynamic systems, and is studied in the present chapter. Control is concerned with the problem of determining a control sequence which brings the system state, in one or more time steps, from the present fuzzy value Xj^ to a desired final fuzzy value Xp This problem is equivalent ot solving the relevant equation Xj-= UjpXjpR with respect to l/^. The fuzzy system identification problem solved here consists in estimating the system fuzzy relational matrix R from a given set of fuzzy input-output pairs (X,-,y,), i= 1,2,...,JV. The no-noise case is considered, since the case where noise is present is stiU a challenging open problem. An algorithm is provided for determining good upper and lower bounds that confine R. Also, the state prediction problem of fuzzy state space models is considered. The chapter closes with a review of three practical fuzzy controllers which have been applied with success to several industrial systems. These controllers are: (i) the fuzzy self-organized controller (SOC), the fuzzy PID supervisor, and the fuzzy PID incremental controller.

2. DETERMINISTIC, STOCHASTIC AND FUZZY SYSTEMS To understand what actually inspired and pushed L-Zadeh to initiate fuzzy system theory through his pioneering article of 1965 [1] one must resort to his earlier studies. In one of them Zadeh introduces the concept of abstract object in order to define what is a system. In his own words «... A system is designed as a collection of ordered pairs, representing inputs and outputs, and defining abstract objects. This way, one may treat various systems as a whole by their causal relationships only, thus dealing with a large variety of problems in a unified way». The next step is of cource to locate an input-output relation with the 69 S.G. Tzafestas and A.N. Venetsanopoulos (eds.) Fuzzy Reasoning in Information, Decision ami Control Systems, 69-96. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.

70

property that it generates input-output pairs characterizing the system. From that point the problem falls within the framework of the classical systemic approach and leads to the development of concepts like the transfer function, frequency response, state space model, etc. Tliroughout the above arguments it was tacitly assumed that the inputs and outputs of the system can be measured quantitatively (nunierically). This is true for technological systems which do not involve the "hum.an" as part of them. However this ceases to be true when the "human" is entered as a component of the system control loop. This is because the human accepts stimulations (inputs) and replies with responses (outputs) that are expressed better by words (e.g. adjectives) than by numbers. The above are best illustrated by the following example, where the human acts as a controller that regulates the proportion of hot and cold water in order to achieve a desjred warm water. For Convenience it is assumed that the flow of water remains constant and only the relative proportion of hot and cold water is changed through a valve. The control rules of the example are: IF "water very hot" THEN 'Valve pretty right" IF "water a little hot" THEN "valve a little right" IF "water warm" THEN "do nothing" IF "water a little cold" THEN 'Valve a little left" IF "water very cold" THEN "valve pretty left" In input - output pair form the above rules are written as: (very hot, pretty right) (a little hot, a little right) (warm, do nothing) (a little cold, a little left) (very hot, pretty left) It is clear that in this case, one cannot find a suitable function that produces the input output pairs, since these pairs are no more numeric but linguistic (here adjectives). According to Zadeh [2] the linguistic description of a system is much more effective, but less specific, than the mathematical (numerical) description. The modern systems theory has been structured and developed around the concept of state space. A system is described by three quantities: input, output and state, (i) If the present state and input determine exactly the system output, then the system is called deterministic, (ii) If the next state and the output (given the present state and input) cannot be described exactly but only through their probabilistic and statistical properties, then the system is called stochastic. (Hi) If for the next state and output (given the present state and input) the only that can be said is that they are elements of a set with fuzzy bounds then the system is called fuzzy. A system acts as a mathematical operator, i.e. as a mathematical function that maps inputs to outputs. In the triad (input, system, output) the system participates with its mathematical model. Thus, in principle, knowing any two of them one can find the third. Fuzzy systems and neural systems are mainly used in estimation problems. They possess an important common advantage over the classical statistical estimation mechanisms and the adaptive control techniques, respectively. Th^y are model free estimators. That is, fuzzy and neural systems can find how an output depends on the input without the need to use a mathematical model. Instead, they use an alternative approach, namely they learn from examples. If X is the input space, Y is the output space, then the pair (Xj, jy) con.stitutes

71

an element of the Cartesian product XxY. In neural systems the sample (Xj, 7,) is a pair of numbers (e.g. (1, 5)), whereas in fu2zy systems this sample is a pair of fu2zy sets (e.g. (fast motion, more green) or (hot water, valve shiightly left)).

3. ABOUT FUZZY RELATIONAL EQUATIONS 3.1 Definition of Fuzzy Relational Equation Let: U, V: Reference supersets F{U), F(V): Sets of all possible fuzzy subsets of U, V respectively UKV: Cartesian product of U, V F{U\V): Set of all possible fuzzy subsets of UxV XGF{U): Fuzzy subset of U YeF{V): Fuzzy subset of V ReF(UxV): Fuzzy relation TTie relational equation among X, Y and R is defined as: y = Xo« where "o" is Zadeh 's compositional rule of inference (max-min composition) [see Ch. 1, Section 9.4]. Using the membership functions n^x), /iy(y) and jU^(jf,y) of X, Y, R respectively the above relational equation takes the form My(y) = max{mm{n

xix),tiif{x,y))

A more general relational equation results if the operator max is replaced by the operator sup and the operator min by a T-norm (see Ch.l, Sec. 5}, namely ^ y ( y ) = sup{^^(Ar)T/ij?(jf,7)} 3.2 Fuzzy System Input - Output Description via Relational Equations A fuzzy system accepts fuzzy sets as inputs and produces other fuzzy sets as outputs. The fuzzy system cannot be described by a classical mathematical model but by a linguistic model, i.e. by a set of rules. Let R be a relation produced by the set of rules that govern the system, X be the fuzzy input set an Y the fuzzy output set. Then an input - output description of the fuzzy system is provided by the relational equation y = XoR For the classical systems the corresponding description is via differential or difference equations. Given the relational equations of a fuzzy system one faces two possible problems: i) Design (Synthesis) Problem: Given Xand y find R, i.e. knowing the input and output, find the system.

72

ii) Measurement Problem: Given Vand R find X, i.e. Knowing the output and the system, find the input. To precisely formulate the above two problems a new operator "«" is required which plays the role of the Inverse T-norm used in Zadeh 's composition rule. This need can be understood by the following argument. From the relational equation Y=XoR one can see that the operator "o" is similar to the operator "•" of the matrbc multiplication. This is so since the operator "o" produces Y (a vector) from the product of X (a vector) and R (a matrix). However, it possesses the fundamental difference that it does not allow the definition of the inverse, i.e. if Y=XoR then it is not true that X= YoR'^. Therefore it is absolutely necessary the introduction of an operator"«" which is the inverse of the Tnorm used in the operator "o". Definition 3.1 An operator (function) «:[0,l]x[0,l]-»[0,l] is defined to possess the properties: i) X " max(y,z) > 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<-' ^k- fuzzy state variable at times k+p, k+p-l,...,k R, ^ fuzzy relations that characterize the dynamics of the system and are defined as: p R : UxXx...xX S : XxY

-

•* [0,1] [0,1]

The first equation is the state equation that combines the fuzzy state at time Jt+p with the fuzzy states at times k+p-l,...,k and the fuzzy control at time k. The second equation is the output at time k+p with the fuzzy state at the same time instant. The above model is of order p and can be considered as a generalization of the difference equation model of order p. In the following discussion it will be assumed that X= Y (the outputs coincide with the states) and that the relation 5 is diagonal. Thus without loss of generality we can work only with the first equation. 4.2 First-Order Fuzzy Systems A first-order fuzzy system is described in state-space by a fuzzy relational equation of the type

74

where Uf^ Xj^ and R have the usual meaning. In terms of membership functions the above equation takes the form ^^k.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<-l Number of Iterations N Step 1: Read the fuzzy input - output sets X, and Yj Step 2: Compute X;« Y, and X;« Y, Step 3: Update the UpperR and LowerR as UpperR*- UpperRriXj« Yj) LowerR'^LowerRu(Xj^ Yj) Step 4: If k
When r= 1 this distance reduces to the well known Hamming distance (Ch. 1, Sec. 6.3). 6.2. State Prediction of Fuzzy Systems The prediction problem of future fuzzy states is stated as follows: Given: The fuzzy system Xj^^j = Xj|,oG and the fuzzy state X^^ at time k. Find: The fuzzy state at time i + m, m > 1. To solve the problem we apply repeatedly the system equation for k=k, k+l, ..., k+m (it is assumed that the fuzzy relation G is known), i.e.

80

^ku.') = ^^*1 °<J = Xi,oG°G = Xu°G

^k*m m Suppose that the fuzzy state at time A^ has yi^ =[0,...,0,1,0,...,0] where the 1 appears at the jth position. Then a normalized measure of the uncertainty of the fuzzy state X^^. j is (Ch. 1, Def. 2.4):

Using n^ =[0,...,0,1,0,...,0] , the fuzzy relational equation X^^j = XjjOG, and the quantity rj, = max/ij^(xy^;), one can compute IXj^ + il and find l^jt.il =[E/^G(^i.^y)]/'-o

Thus the normalized measure of fuzziness of X^^.j is greater than that of Xjj. Similarly the fuzziness measure for X^^.2 is greater than that of Xj^^j, and so on. Thus after a certain number of time instants the prediction is no more useful. 6.3. Illustrative Examples Example 6.1.: Estimation of R Consider the following fuzzy relation

R=

1

0.8

0.9

0.6

0.5

0.7

0 0.7 0.2

0.3 0.9 1

0.5 1 0.7

1 0.8 0.5

0.2 0.6 0.4

0 0 0.2

0 0

0 0.2

0.1 0.3

0.2 0.5

1 0.6

0 1

It is desired to find an estimate of R on the basis of the following fuzzy input - output pairs

81

1

0.3

0.2

0.1

0

0 ]

1

0.8

0.9

0.6

0.5

0.7

0.3

1

0.2

0.1

0

0 ]

0.3

0.3

0.5

1

0.3

0.3

0.2

0.3

1

0.1

0

0 ]

0.7

0.9

1

0.8

0.6

0.2

0.1

0.2

0.3

1

0

0 ]

0.3

1

0.7

0.5

0.4

0.2

0

0.1

0.2

0.3

1

0 ]

0.2

0.3

0.3

0.3

1

0.2

0

0

0.1

0.2 0.3 1 ]

0.2

0.2

0.3

0.5

0.6

1

Applying the upper / lower bound estimation algorithm of Sec. 6 gives the following sequence of fuzzy relations for UpperR and LowerR: 0.8

UpperR

0.9

0.6

0.5

0.7

=

Hamming (t/pper/?,J?) = 17.1

UpperR

=

0.5

0.7

0.5

0.3

0.2

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

0.8

0.9

0.3

0.3

1 1

Hamming {UpperR,R)

UpperR

=

0.6

= 13.8

1

0.8

0.9

0.6

0.5

0.7

0.3

0.3

0.5

1

0.3

0.2

0.7

0.9

1

0.8

0.6

0.2

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

Hamming (L/pperi?,/?) = 11.9

82

UpperR =

1

0.8

0.9

0.6

0.5

0.7

0.3

0.3

0.7

0.9

0.5

1

0.3

0.2

1

0.8

0.6

0.2

0.2

1

0.7

0.5

0.4

0.2

1

1

1

1

1

1

1

1

1

1

1

1

Hamming (L/pper/?,/?) = 9

UpperR

1

0.8

0.9

0.6

0.5

0.7

0.3

0.3

0.5

1

0.3

0.2

0.7

0.9

1

0.8

0.6

0.2

0.2

1

0.7

0.5

0.4

0.2

0.2

0.2

0.3

0.3

1

0.2

1

1

1

1

1

1

Hamming (UpperR,R) = 5.2

UpperR

1

0.8

0.9

0.6

0.5

0.7

0.3

0.3

0.5

1

0.3

0.2

0.7

0.9

1

0.8

0.6

0.2

0.2

1

0.7

0.5

0.4

0.2

0.2

0.2

0.3

0.3

1

0.2

0.2

0.2

0.3

0.5

0.6

1

Hamming {UpperR,R) = 1.9

LowerR =

1

0.8

0.9

0.6

0.5

0.7

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Hamming (R,LowerR,) = 12.9

83

LowerR =

1

0.8

0.9

0.6

0.5

0.7

0

0

0 0 0 0

0 0 0 0

0.5 0 0 0 0

1 0 0

0 0 0 0 0

0 0 0

0 0

0 0

Hamming {R,LowerR,) = 11.4 1 0.8 0 0 0.7 0.9 0 0 0 0 0 0

0.9

0.6

0.5

0.7

0.5 1 0 0 0

1 0.8 0 0 0

0 0.6 0 0 0

0 0 0 0 0

Hamming (R,LowerR,) = 7.4

LowerR =

1 0

0.8 0

0.9 0.5

0.6 1

0.5 0

0.7 0

0.7

0.9

I

0.8

0.6

0

0

1

0.7

0.5

0.4

0

0 0

0 0

0 0

0 0

0 0

0 0

Hamming {R,LowerR,) = 4.8 1

LowerR =

0.5

0.7

0.5

0.6 1

0

0

0.9

0

0.8 0

0.7

0.9

1

0.8

0.6

0

0

1

0.7

0.5

0.4

0

0 0

0

0 0

0

1

0

0

0 0

0

Hamming {R,LowerR,) = 3.8

84

LowerR

=

1

0.8

0.9

0.6

0.5

0.7

0

0

0.5

1

0

0

0.7

0.9

1

0.8

0.6

0

0

1

0.7

0.5

0.4

0

0

0

0

0

1

0

0

0

0

0.5

0.6

1

Hamming {R,LowerR,) =1.7 Hanunlng(UpperR,R)

Hamming(R,LowerR) 20 4-

Fig. 4 Variation of the Hamming distances d{UpperR,R) and d{Lower,R) versus iteration number k

85

Pictorially the sequences of the upper and lower bounds are shown in Fig. 4 where their Hamming distances from R are depicted. Example 6.2: Fuzzy state prediction Consider the system X;t+i=^ik°<^*ith'^Jt = n 0-2 0.1 0 0 0] and

G =

1

0.8

0.6

0.5

0.3

0.1

0.9

1

0.8

0.7

0.4

0.2

0.7

0.9

1

0.8

0.5

0.2 0.7

0.6

0.7

0.9

1

0.8

0.4

0.5

0.7

0.8

1

0.9

0.2

0.3

0.5

0.7

0.8

1

Carrying out the prediction calculations one finds the following sequence of future fuzzy states: Time horizon

Fuzzy state A Sets X 1 i

1

^k*l ° [[1 0.2 0.1 0 0 0]

2

^k*2 = [[1 0.8 0.6 0.7 0.5 0.5]

3

^k.3 = [[1 0.8 0.8 0.8 0.7 0.7]

4

^k*4 ° [[1 0.8 0.8 0.8 0.8 0.7]

5

^k*5 " [[1 0.8 0.8 0.8 0.8 0.8]

6

^k*6 = [[1 0.8 0.8 0.8 0.8 0.8]

Reference superset

Fig. 5 Pictorial representation of the fuzzy states Xj^+i- ^Jt+2> ^Jt+3' ^Jt+4' -^A+5' ^"'^ ^k\6

86 These fuzzy states are represented pictorially in Fig. 5. The size (normalized measure of fuzziness) versus prediction time horizon of the state X^. {k = k, k+ 1, ..., k+6) has the form shown in Fig. 6. "Size" of . fuzzy sets X ' ' 6

•

4 5 6 Prediction time horizon Fig. 6 Size of predicted fuzzy states X versus prediction time horizon

7. THREE SOPHISTICATED FUZZY CONTROLLERS 7.L Self-Organizing Fuzzy Controllers (SOC) This is a fuzzy controller which is an improved version of the basic fuzzy controller and has more intelligency in the sense that it is capable to measure its own performance, and modify the control rules according to a measure of deviation of each output from the trajectory p(kT) where

p(kT) = 0{e{kT),c{kT)}

0)

and 0(e(/c7), c(kT)} represents the performance decision look-up table (relational matrix) used [5]. The structure of the SISO fuzzy SOC has the form of Fig. 7. The performance decision look-up table in [5] is given in Table 1. In this table, zero entries indicate no correction. The further away from desired trajectories the output is, the greater the required control corrections are in one or the other direction. In order to implement this SOC controller which is a kind of fuzzy adaptive cotroller, we need to have an incremental model relating input changes to output changes. For a system with state equation x=f(x,u) the incremental model matrix M is given by: M = TJ

(2)

where / is the system Jacobian matrix J= df/du and T the sampling period. Thus, if an output correction p{kT) is needed, then the required control rule correction is determined

87

e(kT) Hertorinance meausure ., _ ,

incremental r ( k T ) Model

Ruxe mcxiifier Contro] Set p.ointr

GE

i(kT

Output Process -r*

u

TICK^ Gi:

Fig. 7 SISO fuzzy self-organizing control loop by solving the equation p(kT) = Mr(kT) rikT)

as = M-^p{kT)

(3)

The operation of this controller is now obvious from Fig. 7. The performance measure (1) and the incremental model (3) connstitute the higher level of the SOC which coordinates the simple fuzzy controller of the lower level. The control input correction is fed to the rule modifier which modifies the linguistic rules such that future control actions lead to the appropriate output improvement. Some key features of this controller are: The model matrix R is related to the system Jacobian which for SISO systems consist of a single coefficient (usually normalized to unity). For linear MIMO systems the model matrix R involves constant elements (which, if normalized, lie between -1.0 and + 1.0) For nonlinear or nonmonotonic systems the model matrix elements depend on the system state. The incremental model required is much simpler than the exact (possibly nonlinear) model of the system, but it requires the knowledge of the Jacobian J which in many cases is difficult to obtain on-line. To overcome this difficulty one can compute / at the beginning of each control period keeping it constant throughout, and leave to the learning process the balance of any inaccuracies. The controller is always initiated by an initial set of control rules, which are usually crude and are obtained from existing experience. These rules are then iteratively corrected through the rule modification process. The control rule modification procedure involves the following: a) Fuzzification of uncorrected and corrected control input along with the error and change in error, i.e. E{kT-mT) = F{e(kT-mT)} (error)

88 TABLE 1: Performance Measure Look- up Table Change in error e(kT) Towards set-point

Below set-point

Error e(kT)

Above set-point

Away from set -point

-6

-5

-4

-3

-2

-1

0

+1 +2 +3 +4 +5

-6

0

0

0

0

0

0

6

6

6

6

6

6

6

-5

0

0

0

2

2

3

6

6

6

6

6

6

6

-4

0

0

0

2

4

5

6

6

6

6

6

6

6

-3

0

0

0

2

2

3

4

4

4

4

5

5

6

-2

0

0

0

0

0

0

2

2

2

3

4

5

6

-1

0

0

0

0

0

0

1

1

1

2

3

4

5

-0

0

0

0

0

0

0

0

0

0

1

2

3

4

+0

0

0

0

0

0

0

0

0

0

-1

-2

-3

-4

+1

0

0

0

0

0

0

-1

-1

-1

-2

-3

-4

-5

+2

0

0

0

0

0

0

-2

-2

-2

-3

-4

-5

-6

+3

0

0

0

-2

-2

-3

-4

-4

-4

-4

-5

-5

-6

+4

0

0

0

-2

-4

-5

-6

-6

-6

-6

-6

-6

-6

+5

0

0

0

-2

-2

-3

-6

-6

-6

-6

-6

-6

-6

+6

0

0

0

0

0

0

-6

-6

-6

-6

-6

-6

-6

+6

C(kT-mT)=F{c(kT-mT)} (change-in-error) U(kT-mT) = F{u(kT-mT)} (uncorrected control) U{kT-mT) = F{u{kT-mT)+iikT)} (corrected control) where F{-} represents the fuzzification process, mT is the time interval between successive control corrections, and r(kT) is the control correction given by (3). b) Determine the uncorrected and corrected (modified) relation matrices R' and R" from the corresponding implications E{kT-mT) -* C{kT-mT) -* U(kT-mT) E{kT-mT) -* C(kT-mT) •* U{kT-mT) That is R\kT) = E{kT-mT)>^C{kT-mT)y^U{kT-mT) R'(kT) = E(kT-mT)xC(kT-mT)xU'(kT-mT) c) Express the control relation matrix modification linguistically, i.e.

89 R{kT+T)

= {R{kT)

but not R'{kT)

else R"{kT)}

('*)

where R(kT) is the current controller relation matrix and R(kT+ T) is the modified one. In terms of the set operations "n" and "u", equation (4) takes the form: R{kT+T)

= {R{kT)C\~R'(kT)}\JR"ikT)

(^)

The above results for SISO SOC controller can be easily extended to the multivariable case. 7.2. Fuzzy PID Supervisor This type of control was introduced by van Nauta Lemke and Dezhao [6] and consists of a conventional PID controller whose settings are adjusted by a fuzzy supervisor as shown in Fig. 8(a,b). The fuzzy supervisor is designed on the basis of the intuition and knowledge of an experienced human operator such that to improve the performance of the resulting overall supervised PID controller. The architecture of the fuzzy supervisor has the form of Fig. 8b, involving as usual a fuzzifier, a set of fuzzy rules and a deffuzifier. The supervisor accepts the error e and the change-in-error de/dt and adjusts the proportional (P), integral (i) and derivative (D) settings according to the relations P = Po + ^P,

I = ^0+^^' ^ =

^o*^^

where P^, /„ and D^ are re the initial settings of the controller, and AP, Af AI and AD are their adjustments (corrections). ins). As shown in Fig. 8b five scaling factos (i(gains) are used in the supervision namely K^, K^ , Kp, Kj and Kjj which are defined by

AP = KpPf, AI = Kjlf,

AD = K^Df

with |r|=0.1 for-0.1
m

/=!

1=1

where Vy is the geometric center of the component lif^{v) in the membership function m 1=1

90

(a)

e

1

^^©^16—P

K i. 'uzzifier

De fuzziFier

_e__

(b) Fig. 8 (a) Supervised PID controller, (b) The architecture of the supervisor Other defuzzification rules that en be used are the "center-of-gravity" rule and the "equal-surface" rule [6]. The performnace index is in line with the human judgment, and uses the rise time T^ the overshoot h and the ITAE criterion to qualify the overall performance of the controlled system as bad, satisfactory, good and excellent. The selected fuzzy subsets for T^ h, ITAE, and the overall performance index (P/) are shown in Fig. 9 from which the following linguistic rules are obtained:

0

10

15

20 25 ITAE {V.sec^}

2

3 4 5 Performance Index

Fig. 9 Fuzzy subsets for h, ITAE, T^ and performance index

P.I.

Rl: R2: R3: R4: R5: R6: R7:

Jf h is big OR ITAE is big THEN the performance is bad If h is medium AND ITAE is medium THEN the performance is satisfactory If /j is medium AND ITAE is small AND THEN the performance is satisfactory If /i is small AND /TA£ is medium AND THEN the performance is good If h is medium AND /TAH is smaW AND THEN the performance is good If h is small AND ITAE is medium AND THEN the performance is good If b is sma/i AND ITAE is smaii THEN the performance is excellent

T, is big T^ is NOT big T^ is NOT big T^ is NOT b/g

As usual, the performance index is computed by the fuzzy cartesian product of the above linguistic rules and the weighted sum defuzzifier rule. 7.3. Fuzzy PID Incremental Controller This controller [7] is based on the same basic philosophy as the fuzzy PID supervisor presented in Sec. 7.2. The difference is that the controller parameter changes are determined using MacVicar-Whelan 's table [8]. The PID controller settings are initially Slx-S

-0

+0

+ S +M +1.

+S -M

-L

-L

-u

-1.

-L

-0

-S

-M

-M

-M

-u

-L

+S +M

+s

-0

-S

-S

-S

-M

-U

+ 0 4M

•IM

+s

+0

-0

-S

-M

-M

-0

-fM

+M

+s

+0

-0

-s

-M

-M

-s

+L

>M

+s

•t-S

+s

+0

-5

-s

-M

+L

+L

+M

4U

+M

+s

+0

+0

-L

+L +L +L

+u

+L

-fM

+0

+0

H.

+ u iO •fM

+s

-M

Jf*o

Fig. 10 MacVicar-Whelan fuzzy matrix

92 determined by some standard nonfuzzy technique such as the Ziegler-Nichols technique, the analytical technique, the Kalman digital technique, etc. [7]. The idea is to slightly change these settings during the system transient so as to improve the step response characteristics. MacVicar-Whelan [8] has observed that the fuzzy control matrices of King and Mamdani [9] were not complete. There are situations where the action to be taken is not defined. That is, there is not always agreement about just what value the dependent variable should take for a given value of the independent variable (i.e. the error) and its derivative. The fuzzy control matrix proposed by MacVicar -Whelan is shown in Fig. 10 and is based on the Following principles: a) If the output has the desired value and the error derivative is zero, then we keep constant the output of the controller. b) If the output diverges from the desired value, then our action depends on the signum and the value of the error and its derivative. If the conditions are such that the error can be corrected quickly by itself, then we keep the controller output constant or almost constant. Otherwise we change the controller output to achieve satisfactory results. Most human controllers act by folowing the philosophy of this matrix. The logic of this matrix approaches the human logic more closely than other fuzzy control matrices. This matrix also shows something very fundamental. The designer, prior to designing a fuzzy controller, should study the actions of the human controller, in order to be able to give to the fuzzy controller a more realistic behavior. Note that if we project the policy of the matrix on an axis perpendicular to the main diagonal, then we will see clearly the form of the policy. One can increase the resolution of the matrix by a factor of 2. This increases

Fig. 11 Increasing the control resolution by a factor of 2

93

the number of the projected parts and reduces the fuzziness of the control policy (Fig. 11). The analysis of the projections of the boundaries can be regarded as a study of the variance of the neutral point and the amplitude of the fuzzy function /u^. MacVicar-Whelan has asserted that using this analysis one can predict the resolution of the projections of the boundaries. An alternative solution would be to start with a small resolution and slowly increase it until we get the desired performance. This matrix was not used by others, probably because it implements the fuzzy controller in a direct way, and so for each particular case with special requirements it should be varied. Another drawback is the small number of quantization levels used. However, we believe that the logic and structure of this matrix are valuable, and thus we used it for improving existing analog and digital PID tuning procedures as described in the following. The rules used for shlightly changing the controller parameter settings during transient are empirical and are involved in a fuzzy 14x14 control matrix. Th eproblems which must be faced refer to stability, computation time, cost, quantization, and generality of the approach. The main issues of our control rules are as follows: a) since the integral term is responsible for the overshoot, shlightly decreasing it at the moment when the system response exceeds the value 1, one can reduce considerably the overshoot. On the other hand, a small increase of the integral term during the rising of the response, leads to a 10-20% improvement of the rise time.

-6

-5

1 ""*1-^ -1 !

-6 -5 -4 -3 -2 -1 -0 +0 +1

+2 +3 +4 +5 +6

-6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0

-6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0 0

-5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0 0 +1

•

+0

1-1 +2

-3 -3 -3 -2 -2 -2 -2 -1 -1 -1 -1 0 0 0 0 0 0 +1

-2 -2 -2 -1 -1 -1 -1 0 0 0 0 0 0 +1

-0

'

-5 -4 -4 -3 -3 -2 -2 -1 -I 0 0 0 +1 +1

-4 -4 -3 -3 -2 -2 -1 -1 0 0 0

-4 -3 -3 -2 -2 -1 -1 0 0 0

+ 3 + 4 +5

+6

0 0 0 +1 +1 +2 +2 +3 +3 +4 +4 +5 +5 +6

0 0 +1 +1 +2 +2 +3 +3 +4 +4 +5 +5 +6 +6

-1 -1 -1 0 0 0 0 0 0 +1 +1 +1 +1 +2

+ 1 + 1 +2 +2 + 1 +2 +2 +3

+ 1 + 1 +2 + 2 + 3 + 3

+ 1 + 1 +2

+ 1 + 1 +2

+ 1 +2 +2 +3 +2 +2 +3

+2 +3 +3 +4

+ 2 + 3 + 3 +4 + 4 + 3 + 4 +4 + 5

+ 3 + 4 +4 + 5 + 5

Fig. 12 The present fuzzy PID control matrix

94

b) Since the derivative term is responsible for the flatness of the step response, a small increase of it during rising and in the steady state eliminates the small oscillations that usually occur. c) Increasing the proportional term leads to reduced rise time and increased oscillations. Thus finally this term should be decreased to avoid oscillatory behavior. All these issues are experential observations of human controllers. Our problem was to accomodate them into a matrbc and find the criteria of actually carrying out these changes. The amount of increase / decrease of the proportional, integral and derivative terms is expressed by three coefficients k^ k^ and k^ The fuzzy PID control matrix is shown in Fig. 12. The basis for the control rules is the error E=X^-X {X^ is the desired and X the actual system output) and its derivative AE=A(Xg-X). The sampling of E and AE should be made at small time interavals of the order 0.1-0.2 sec. This table involves 14 quantization levels for both E and AE. This ensures a closer follow-up of the unit step response. The linguistic code is + •» positive change - — negative change 6 •• extra large 4 •• big 1 — extra small 5 •• large 3 •» medium 0 •» zero (null) 2 •• small The changes of the three terms, which showed good success, are given by the expressions: P = P*CV{E^E}

X ^1

(Proportionar)

I = I + CV{EAE}

X k2

(Integral)

D = D + CV{EAE}

X k^

(13)

(Derivative)

The parameters k^, k2 and k^ play an important role in the whole procedure, since they determine the range of variation of each term. For example, if some tuning method ensures very small rise time and large overshoot, then the integral term should have a large range of variation, whereas the other terms can remain unchanged. This range of variation should be matched with the stability interval in order to guarrantee stability. Thus the values of Jtj, k2 and k-^ are determined from both the stability analysis and the particular characteristics of the closed-loop response. In general the parameters k^ /cjand k-^ provide large flexibility and can be used in conjunction with all available PID tuning algorithms. We have also used the fuzzy control matrix of Fig. 13. The only difference with the matrix of Fig. 12 is that here the Neperian logarithm of \CV{E, AE] \ is used. This ensures greater stability around the set point and smoother transition from one value to another. The results obtained using this table are also satisfactory, but the values of /cj, Aj and k^ are found to be larger. The stability can be tested by various techniques. In our studies we used Routh 's table in the continuous-time case and the corresponding table, obtained through the bilinear transformation z=(w+l)/(w-l) in the discrete-time case. The most serious problem faced in our experiments was that of quantization. The right quantization of E and AE ensures a good evaluation of the controller performance. We found the logarithmic quantization to be most successful. We have used the following quantization levels:

95 ^E{X„-X) ^ -6 -5 -4 -3 -2 -1 -0 +0 +1 +2 +3 +4 +5 +6

-1.79 -1.79 -1.60 -1.60 -1.38 -1.38 -1.10 -1.10

• '

-1.79 -1.60 -1.60 -1.38 -138

-

-^

-2

-1.60 -1.60 -1.38 -1.60 -1.38 -1.38 -1.38 -1.38 -1.10 -1.38 -1.10 -1.10

-1.10 -1.10 -1.10 -1.10 -0,69 -0.69 -0.69 -0.69 -0.69 0.00 -0.69 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

-1.10 -0.69 -0.69 -0.69 -0.69 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00

+0

+4

^'

+6

0.00 0.00 0.00

0.00 0.00 0.00

0.00

0.00 0.00 0.00 0.00 + 0.69 +0.69

0.00 0.00 0.00 + 0.69 0.69 + 1.10

0.00 0.00 0.00 0.00

+ 1.10 + 1.10 + 1.10 + 138

+ 0.69 + 1.10 + 1.10 + 1.10 + 1.10 + 1.38 + 1.38 + 1.38 + 138 + 1.38 + 1.60

+ 1.10 + 138 + 1.38 + 1.60 + 1.60

+ 138 + 1.38 + 1.60 + 1.60 + 1.79

-

-

-1.38 -1.10 -1.10

-1.10 -1.10 -0.69

-1.10 -0.69 -0.69

-0.69 -0.69 0.00

-0.69 -0.69

-0.69 0.00

0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 +0.69 0.00 +0.69 + 0.69 +0.69 +0.69 + 1.10 +0.69 +0.69 + 1.10 + 1.10

"

" -0.69 0.00 0.00 0.00 0.00 0.00 0.00 0.00 +0.69

-

+ 0.69 +0.69 + 1.10 + 1.10

0.00 0.00 0.00

0,00 +0.69 +0.69 +0.69 + 0.69 + 1.10

+ 138 + 1.60 + 1.60 + 1.79 + 1.79

Fig. 13 An alternative fuzzy PID control matrix O^a^=0.02, DQ^^^O.02. Let us see in detail the logarithmic quantization. If the error E is greater than O^^x/^' ^^^'^ ^' ^^ quantized in a value between + 1 and +6. l{E<-0^^/2, then it is quantixed in the interval -6, -1. If 0<E
96 analysis, stability analysis, sensitivity analysis, parameter and state estimation, and controller design. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

L.A. Zadeh: Fuzzy Sets, Inform, and Control, Vol. 8, pp. 338-353, 1965. L.A. Zadeh: Outline of a New Approach to the Analysis of Complex Systems and Decision Processes, IEEE Trans. Syst. Man, Cybern., Vol. SMC-3, pp. 28-44, 1973. W. Pedrycz: An Approach to the Analysis of Fuzzy Sytems, Int. J. Control, Vol. 34, No.3, pp. 403-421, 1981. E. Sanchez: Eigen fuzzy Sets and Fuzzy Relations, J. Math. Anal. Appl., Vol. 81, pp. 399-421, 1981. E.H. Mamdani, and TJ. Procyk: A linguistic self-organizing process controller, Automatica, Vol.15, pp. 15-30, 1979. H.R. Van Nauta Lemke, and W. De-Zhao: Fuzzy PID supervisor, 24th IEEE CDC. Paper WP9-5:15, pp. 602-608. Ft. Lauderdale, FL. 1985. S.G. Tzafestas, and N. Papanikolopoulos: Intelligent PID control based on fuzzy logic, Proc. IFAC Symp. on Distributed Intelligence Systems Methods and Applications (DIS '88), Varna - Bulgaria, June 1988. P.J. MacVicar-Whelan: Fuzzy sets for man-machine interaction, Intl. J. ManMachine Studies, Vol. 8, 687-697, 1976, P.J. King and E.H. Mamdani: The application of fuzzy control systems to industrial processes, Proc 6th IFAC World Congress, Boston 1975, Automatica, Vol 13, pp. 235-242, 1977.

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PART 2

NEURO-FUZZY SYSTEMS

CHAPTER 5

NEURAL NET APPLICATIONS IN FUZZY SYSTEMS Hiroshi Kavvamura and Akinori Tani Department of Architecture and Civil Engineering Faculty of Engineering, Kobe University Rokkodai, Nada, Kobe, 657, Japan

1. Introduction The concepts of fiizzy sets and fuzzy systems proposed by Zadeh [18] [19] have many possibilities of being applied to the description and analysis of this real fiizzy world. In the application of these concepts to knowledge engineering, there are two problems, i.e., learning of fuzzy knowledges and automatic inference of unknown variables. On the other hand, one can identify any complex functions by means of perceptron-type neural networks [12] and the back-propagation error method [13]. The purpose of this chapter is to propose some methods of neural network applications to fuzzy systems (and not of fuzzy system applications to neural networks [10]). As mentioned above, the concept of neural networks is very useful for learning knowledge and automatic inference. In this chapter, besides usual neural networks, a couple of quasi-neural network methods are proposed. In the field of engineering planning and design, evaluation and optimization are performed by using many kinds of variables (e.g., load, size, weight, strength, intensity, deformation, cost and so on) and constraints (e.g., design formula, mathematical equation, function, code, criterion and so on). Due to inevitable uncertainties, in general, these variables and constraints can be described with fuzzy sets and fuzzy relations [17][18], respectively, which can include rationally usual crisp, deterministic and probabilistic expressions as special states. As conditioned probabilities are used in probability systems, so conditioned fuzzy sets [1] are used as states in fuzzy systems composed of inputs, states and outputs [19]. In engineering problems, such conditioned fuzzy sets which belong to fuzzy relations are very useful enough to discriminate uncertainties in variables firom ones in constraints (or functions) [3]. The final purpose of this paper is to propose a paradigm of intelligent and neural fuzzy networks in which learning, identification, evaluation and optimization can be performed for engineering planning and design by expanding fiizzy systems and employing a couple of models similar to neural networks [10]. 99

S.G. Tiafestas and A.N. Venetsanopoulos (eds.) Fuzzy Reasoning in Information, Decision and Control Systems, 99-119. © 1994 Ktuwer Academic Publishers. Printed in the Netherlands.

100

2. Identification of States in Fuzzy Systems 2-1. Fuzzy System Fuzzy systems arc composed of input X (fuzzy set), state R (fuzzy relation) and output Y (fuzzy set) as shown in Fig.l [19]. If R is described with conditioned fuzzy sets R(Y/X), the system shown in Fig.l becomes directed (solid allows).

Input

State Output Fig.l Fuzzy System

The inference algorithm is given by XoR(Y/X) = Y,

(1)

where o means the max-min operation [18]. If R is assumed to be described with an undirected fuzzy relation, i.e., R(Y/X)=R(XA'),

(2)

the system becomes undirected (solid and broken allows). For the purpose of the maxmin operation in fuzzy systems, fuzzy relations are described with matrix expression. Consider a two-dimensional matrix fj. ^ (0 ^ /ij^^ I) as shown in Fig.2, and this matrix means a state (i.e., fuzzy relation) R. If R is valid for both the cases of given x and given y, the undirected fuzzy system can be illustrated without allows as shown in Fig.3. y

y 1

y 2

3

Fig.2 State (Fuzzy Relation) Described with Matrix Expression

Fig.3 Undirected Fuzzy System

If fiizzy relation matrices are identified by using appropriate fiinctions, such undirected fuzzy systems can be constituted, bi this chapter, neural network, fuzzy confluence rule and fuzzy identifier are employed for the identification of states in fuzzy systems. The second and third functions were proposed by the authors [3] [7] for the identification of engineering planning and evaluation functions. The outlines of these three functions and the similarities between them will be explained in the following sections.

101

2-2. Neural Network A typical neural network is shown in Fig.4 [13]. Input-, hidden- and output-layer unit values, Oj, Oj and oj^, are given by aj-Xj,

bj=f(Uj), Ck=f(Uk),

i=l,....,ni. j=l,«"«,nj,

(3) (4) (5)

k=l,"",n]{,

" 1

uj=Ewjj«aj+(9j,

(6)

i=l

"J Uk=Zwkj'bk+0k'

(7)

k=l

where x is input, w is connection weight and 6 is threshold value. function expressed by Eq.8 is often used as the function f(u) in Eqs.4, 5.

The sigmoid

f(u)=l/{l+exp(-u)}

(8) Output Valuex ^

%(x,y) = R ^

Hidden Layer ^ Unit Value .b j

Neural Network

Input Value:a j X

Fig.4 Neural Network

y

Fig.5 Identification of R by Neural Network

If x j , X2 and c^ in Eqs.3~7 are replaced by x, y and /j. ^ respectively, the function of neural network can be illustrated as shown in Fig.5. U R is membership value of fuzzy relation as shown in Fig.2, which implies that we can identify fuzzy relation R by using the neural network (e.g., the back-propagation error method [13]) and that after the identification we can constitute a fuzzy system with the state R such as shown in Fig.3. 2-3. Fuzzy Confluence Rule [3] [4] Even in engineering planning, sometimes we have to evaluate subjectively numerical values and/or subjective judgements. For the purpose of such a subjective evaluation, the authors already proposed a fuzzy confluence rule [3] expressed by a linear combination of intersection-, union-, algebraic summation- and product-membership

102

functions n with mutual influence coefficient a , over estimation coefficient ^ and weighting coefficient 7 as follows: ; t i R = ( l - a X l - y 9 ) j " i + a ( l - / 9 ) / z p + ( l - a ) ^ Mu+a ^ ;"s

(9)

in which Hl=!\{UXi(xi)} 7 i, Ui?=n{liXi(xi)} f i, i=l i=l n n H\j=\jy\H Xi(xi)> /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
(12)

0^a,yS,ri^l,(i=1.2,..-,n)

(13)

(11)

The topological effects of the coefficients a and ^ on the linear combination are illustrated in Fig.6. This fuzzy confluence rule can be illustrated as a kind of neural network with hidden-layer units, u^, jUp, fi^jSni ^ g such as shown in Fig.7. /ZR(x,y)^R 10

Fuzzy Confluence Rule

©

-"xW

Ax^^a* '"X^'^s'

J"xW

4^ X

y

Fig.6 Topological Effect of Fig.7 Fuzzy Confluence Rule Fig.8 Identification of R by a and /S as a kind of Neural Network Fuzzy Confluence Rule The identification (i.e., learning) of this rule can be performed by determining a, jS, 7 1,' • • , 7 n based on given sample data. The method of least squares is effective for the identification [4]. This rule has a great merit, i.e., "not black box", which is different from usual neural networks. If XI and X2 in Fig.7 are replaced by x and y, the function of fuzzy confluence value can be illustrated as shown in Fig.8. After the identification of R, we can constitute a fuzzy system with the state R such as shown in Fig.3. 2-4. Fuzzy Identifier [7] In engineering planning and design, when we search for such an optimal solution of systems as it needs plenty of time or money to calculate, we have to predict it based on

103 limited results of analysis. Aiming at solving these problems, the authors already proposed a multi-variable fuzzy identifier shown in Fig.9 [7]. Multi-variable Fuzzy Identifier c=> y=f(xi,)fe,..,x„) : ''n-^ nO ^0 Tidentification Output of Inputs of lata Sets Initial Data Fuzzy Set Numerical Values Fig.9 Multi-variable Fuzzy Identifier ' ' l - ^ lOi

Xf=a2o

If y=f(xi, X2,"",Xn) is assumed to be a linear equation expressed by (14)

y= a 1X1+ a 2x2+- • • + a nXn+ a n+h using n+1 sets of initial data, i.e.,

(15)

(xi=aii, X2=a2i,"", Xn=ani, y=bi), i = l , " " n + l ,

the coefficients of Eq.l4, a 1, Q ; 2 ' ' " ' '^n+l ^^ ^ determined by solving simultaneous linear equations. When the following point in the n-dimensional space is given as an input, (16)

xi=aio, X2=a20,-", Xn=anO the output yo is obtained by

(17)

yO= « 1^10+ a 2320+" • •+ « nanO+ « n+1

Here, let the minimum b and the maximum b among bj, i=l,«.-,n, in Eq.l5 be yomin and yomax' respectively, and one can illustrate a triangle-type membership function [yOmin. yO' yOmaxl ^ shown in Fig.lO.

1-

0 .y"^^

A^^bi

^^%^

y \ 1.

"^ >

yOlmin Vol yfllmax ^0 V Fig.lO Average Membership Function

If a couple of groups of appropriate n+1 initial data sets (Eq.l5), i.e., three points with the first, the second and the third shortest distances from the input point (Eq.l6) are employed, membership functions [j. YQ\, fJ- YO2' ^ Y03 ^^'^ ^^^^^ average membership

104

function fx yo' '•^•» * ^ ^"^' output of the proposed fuzzy identifier are obtained as shown in Fig. 10, where average means algebraic one with respect to membership values. This multi-variable fuzzy identification can be illustrated as a kind of neural network with two hidden-layer units, i.e., linear equation and avarage as shown in Fig.ll. The authors already applied it to structural planning [2].

(J-yp^ R(Y/X)

Avarage Linear Equations

1

Fuzzy Identifier

f3 ^1

^2 ^3 ''n a 10 320 ^30 ^nO Fig.ll Multi-variable Fuzzy Identifier as a kind of Neural Network m

,

t Fuzzy Identifier

y (a) Input X (b) Input y Fig.l2 Identification of R by Fuzzy Identifier

The simplest fuzzy identifier with input x is shown in Fig.l2(a), where output fi y/x is conditioned membership function and corresponds to conditioned fuzzy set R(XA'). On the contrary, when y is put into the fuzzy identifier, /i ^ A ' ^"'^ RCXA') can be obtained. Therefore, the fuzzy identifier can be regarded as undirected fuzzy system as shown in Fig.3. 2-5. Calculation Examples 2 - 5 - 1 . Neural Network Consider a fuzzy system as shown in Fig. 12 in which X and Y mean structural damage and return period of earthquake loading, and fuzzy relation R is defined as a label "structural safety". The membership function fi ^ shows the grade of structural safety, fi j{(x) and (i Y(y) are transformation functions into R and simultaneously the safety membership functions of x and y. X:StnJC1ural Damage

&

/ixW

R:Structura] Safely

-{j^vXlix^T)

Y:Relum Period of Earthquake Loading

l^yCy)

-tu

Fig. 12 Fuzzy System for Structural Safety Evaluation The structural safety evaluation is based essentially on subjective judgements and can be determined by questionnaires regarding school buildings distributed to structural experts [15]. The results of questionnaires arc plotted in Figs.l3, 14 and 15, which show U x(x) of structural damage x, ij. yCy) of return period of earthquake loading y and p. R with respect to // j^ and ii y, respectively.

105

Solid

Collapse

Fig. 13 Saftey Evaluation of Structural Damage by Questionnaires

0

20

40

60

80

IC,

Fig.l4 Safety Evaluation of Return Period of Eairtliqualce Loading in Design by Questionnaires

To identify ^ ^ in Fig. 15 by neural network, a neural model is assumed as shown in Fig.l6 [16]. The connection weights w and threshold values 0 are determined by the back-propagation error method [13] as shown in Table 1. The final output /i R is plotted in Fig.l7 wliich shows very nice agreements to Fig.lS.

^ Y

"•"

1

Fig.lS Safety Evaiuation with Respect to // x and |i y by Questionnaires Output /i J

&i\A 4<"23f31 Z\%lJ\j wKaWi

hm llf21''Or^ Ills X ^ Input- n A

m

-yi^!2%

./'•

0

In p«t-/f^(y)

Fig. 16 Assumed Neural Network

Fig. 17 Safety Evaluation with Respect to IIX and /i Y by Neural Network

106

Tabic 1 Identification of Q and w by Neural Net^'ork Biases Wei ghts 1.02 -3.27 1.05 ^31 2l"*'31 13^21 1.36 -1.76 -2.58 ^21 22^31 14^21 -1,81 4.82 1.58 6*22 23^31 11^22 -2.08 -7.96 3.22 ^23 24 "^Sl 12*^22 -1.31 -0.939 -1.98 6" 24 11*'21 13*22 -1.07 2.15 12^21 t4*'22 2-5-2. Fuzzy Confluence Rule To identify n ^'m. Fig. 15, a fuzzy confluence rule model is assumed as shown in Fig.18 [16]. The paramctcis, a , jS and j are determined by the method of least squares and showH in Table 2. The final output /^ g is plotted in Fig.l9 which sliows very nice agreements to Fig.15.

Output /ig

Input-/ixCx) Input-/I YCJ)

H Y"

--

i

Fig.18 Assumed Fuzzy Confluence Rule Fig.l9 Safety Evaluation with Respect to fi X and /I Y by Fuzzy Confluence Rule Table 2 Identification of a , /3 and j by Fuzzy Confluence Rule a /S Ti T2 F. C. Parameter 1.0 0.48 1.0 0.2 2-5-3. Fuzzy Identifier The fuzzy identifier shown in Fig. 11 can present the optimal value yo whose membership value is the maximum in /£ yo- Therefore, it enables us to examine its exactness as a function [7]. Assume a function given by Eq.l8 y=(xl/x2)+x3

(18)

107

shown in Table 3 in which b j (i=l, anda3i(i=l,»»»,5).

^1

Hi

^2

^21

% y

%i

1,5) is derived from Eq.18 and assumed an, a2i

Tabic 3 Initial Data Sets 1 2 3 4 1 4 3 5 2 3 5 8 2 4 0.5 8 4.5

h

3

1.3

5 7 10 12

8.625

12.7

Table 4 1331 Combination Cases of Input Data for ExaminatioB l~7(pitch:0.6) Xl %! 2~10(pitch:0.8) ^2 ^2 0.5~12(pitch:1.15) ^^3 %i f

1

•

•

t

• "'

*

"'"""'

—11

1331 Cases By using the fuzzy identifier model shown in Fig.ll and the initial data sets shown in Table 3, the final optimal values of y arc estimated in the 1331 combination cases of input data shown in Table 4. The comparison between the calculated values y^al by fuzzy identifier and the real values Jfc-^i by Eq.18 is shown in Fig.20. In this case study, they coincide well with each other. 17.5 ^real

12.5

ycal 17.5

Fig.20 Comparison between Real Vaie by y=(xl/x2)+x3 and Calculated ones by Fuzzy Identifier 3. ConstitutioB of Intelligent Fuzzy Networks 3 - 1 . Fuzzy Networks In engineering planning and design, X and Y are considered to be variables described with fiizzy sets and R is a constraint described with a fuzzy relation. When undirected fii77.v R1/Stf»,Tn.'5 arp niTinp,rtp.d -with p.ar.h n t h e r a fiiyTir nphiyriTlf ic crtnstmftfA

nc chnm/n In

108 Fig.21 in which X^, • • • ,Xjj and Rj, • • • ,R^ are variables and constraints, respectively, and inference can be performed in arbitrary directions.

Fig.21 Fuzzy Network For example, a simple fuzzy network is shown in Fig.22 by which a short column subjected to axial load P can be designed, a g^is axial stress, S is safety factor and S^ is critical safety factor. The fuzzy relations P/A o a and S ^ S^. mean stress and safety evaluations, respectively. P(load) {}

A(Area)

CTa(Stress)

Fig.22 Stub Column Design Network 3-2. General Methods for Fuzzy Network Reasoning Now, let's go back to fuzzy networks as shown in Fig.21 where states R^, R2,»'> and Rjjj are identified by learning given initial data. The next subject is to obtain necessary or unknown variables Xj, Xj, etc. by fuzzy network reasoning. The most perfect method is to get the total state Rj whose variables X]^,» • ^Xji are all unknown as shown in Fig.23.

R,= n Rv

Fig.23 Total Fuzzy System with Unknown Variables The optimal solution is given by a vector (x^*, X2*,—, x^*) whose membership value is maximal. Before getting a total state Rj, known or given variables have to be eliminated by inserting them to connected states by the max-min operation. However, this total method seems to be very difficult because generally the number of variables is very large.

109

A more approximate method is successive shadow one in which even unnecessary variables are eliminated successively. This method is similar to the solving multivariable simultaneous linear equations. The elimination can be performed by using the max-min compositional rule [18]. For example, fuzzy relations R X Y ( X , Y ) and R Y Z ( Y , Z ) can be connected and reduced to Rxz(X,Z) as follows: (19)

RXY(X,Y)ORYZ(Y,Z)=RXZ(X,Z),

The procedure of Eq.l9 is illustrated with fuzzy network as shown in Fig.24.

Fig.24 Elimination of Y

Fig.25 Final Fuzzy System

When the number of variables X is equal to the number of states R, finally the network is reduced to a fuzzy set such as shown in Fig.25 which is a solution itself. In general, the composition rule gives shadows of membership functions fl8]. So, the membership function of the final fuzzy set shown in Fig.25 is the shadow of the membership function of the total state Rj shown in Fig.23 projected to Xj-plane. 3-3. Intelligent Fuzzy Network Reasoning 3 - 3 - 1 . Frame Knowledge Representation Method [6] In this section, two kinds of practical and intelligent solving methods of fuzzy network are shown aiming at object-oriented systems. The first is frame knowledge representation method [6]. Table 5 shows common frame knowledge representations, i.e.. Frame 10 for inputs and outputs and Frame S for states of fuzzy systems. In these frames, "knowledge" represents fuzzy set, and D shows input, state or output in sub-frames such as shown in Table 6. For Example, if fuzzy systems A and C as shown in Fig.26, sub-frames, i.e., Frame IB, SB and OB are shown in Table 6. Actually, sub-frames have to be constructed according to fuzzy networks such as shown in Fig.21.

lA SA OA Fuzzy System A

IB SB OB Fuzzy System B Fig.26 Serial Fuzzy Network

]C

sc oc

Fuzzy System C

110

By using Tables 5 and 6, when a certain variable or state is denoted as a final goal, the most terminal variables or states are searched out (backward inference). When the most terminal variables or states are given, the goal is obtained (forward inference). Table 5 Common Frame Knowledge Representation Naie of Fraie Knoiledge Procedure

Fr arae

10

Frame S HQ (= O )

» • (=[•]) 1. If knoiledge D ] exists then exit tliis fraie. 2.!ecognilioii of fra»e naiie ihich calls this fraje. 3.1f G is not called by D tiencall Q 4.1f D is not called b j D then call Q 5.If n exists and D is not called by D then call D and put [D] into O 6. If D does not exist or D is not called by D then input [Q] and exit this fraie. 7. If knoiledge D ] doesn't exist or D is not called by D or D then D ] = D ] and exit this fraie. 8. Calculation of D ] using O and O 9. If knoiledge D ] and D ] exist then 0 = i i n l O . O } and exit this frai&

1. If knoiledge D ] exists then exit this fraie. 2. Secognition of fraie naie jhich calls this fraie. 3. If D is called by • or D then input O and exit this fraie. 4. If D is not called by D then call • . 5. If D is not called by D then call D. e. Calculation of D ] using O and D ] and exit this fane. n

shots naie of fraie.

O

shots naie of knoiledge.

Table 6 Sub-Frame Knowledge Representation For System B Naie of Fraie

Frame IB

F r a m e OB

Frame

ISA

iFrame 10|

1 Frame

iFrame

Knoiledge

DDQB]

H®

HQB]

Procedure

1.E

1. H 3. [¥1 [¥] QB]

1. QB]

3.

QB]

[i]®

10 |

7. OF] QB] H ®

QB]

QT]7. QO H QE B H

8. Q B ] Q B ] | 0 B ) ( O operation)

J. \^

9. [¥] n n

s. QB] H

QBI QB]

QFl

4. QB] QB] QB] s. QB] [¥] QB] 8. QB] QB] QB] »

6. [c] QC] @ [OB]

QB] QB]

S|

3. QB] | ¥ ] QB] d ]

4. [ H QB] QB] 4. QB] [U H] s. [T] [¥] H E H H 5. H] [D Qc] Qi] QC] H ] E. [ A ] H

SB

H •(a

operation)

[§] IjlJ (tax-tin operation)

QB] H

QD

3-3-2. Inference Tree Method [8] [9] The next is inference tree method [8] [9]. When a certain variable is denoted as a final goal in a fuzzy network such as shown in Fig.21, a hierarchical tree with the top event of the final goal is constructed and the most terminal variables are pointed out as shown in Fig.27 according to given research rules (backward inference). The research

Ill

algorithm is shown in Fig.28. When the most terminal variables are given, the goal is obtained according to the max-min operation (forward inference).

Tenninal

X^_i

Fig.27 Fuzzy Hierarchy with Top Event X i START

-»| Depth of Hierarchy :1| Search states iS which are related to outputs iO in the i-th hierarchy level

3ZI

Knowledge base (KBi) with respect to iS and iO is created. 0 I iOl , i02. •••*. iOn

iOj is 'Input'

iOj is deternined. iOj and iSa are deleted from KBi. One of the outputs which related to iSn is needed demanded as an ' Output' and then another outputs ' Input'.

Fig.28 Reasoning Algorithm

are to be by users, become

112

The first method is not substantially different from the second one. In the first method, however, the variables and states are not supposed to be given. So, it is possible to denote even a state as a goal and even states can be pointed out as necessary inputs. Only when all the states are given from the beginning, the second method becomes usefUl. 3-4. Inference Examples 3 - 4 - 1 . Frame Knowledge Representation Method [6] Fig.29 shows an intelligent fuzzy network for aseismic structural evaluation and optimum design in which directed fuzzy systems are employed. R, V, D and S means the fuzzy sets which represent the likelihood of earthquake return period, maximum ground velocity amplitude, seismic structural damage and structural safety, respectively. V/R, DA' and S/D/R mean conditioned fuzzy sets which represent the likelihood of seismicity, aseismic structural characteristic and criterion of aseismic structural safety, respectively. S/D/R is described with non-fuzzy parameters D and R and given from structural, economical and social points of view. A

B S~

o

^/R

D, 7V

> D

> V <-^ V

^ ^/D/R R ;Retum Period of Seismic Load V :Ve]ocily Amplitude of Ground Motion D ;Damage to Structure

'^ R ^

S :Safety

Fig.29 Intelligent Fuzzy Network for Aseismic Structural Evaluation To evaluate aseismic structural safety, aseismic structural safety S is called initially. Calling and reasoning flows (hollow and solid arrows) in the case of calling structural safety are shown in Fig.30. According to the frame knowledge representations shown by Tables 5 and 6, R, V/R, D/V and S/D/R are demanded of users. When the fuzzy sets shown in the dottcd-line rectangles in Fig.30 are put into this network, the aseismic structural safety S shown in the double-line rectangle is finally obtained. When aseismic structural characteristic D/V is called initially, one can perform optimum aseismic structural design. Calling and reasoning flows (hollow and solid arrows) in the case of calling D/V are shown in Fig.31. According to the frame knowledge representations shown by Tables 5 and 6, R, V/R, S/D/R and S are demanded by users. When the fuzzy sets shown in the dottcd-line rectangles in Fig.31 are put into the network, the aseismic structural characteristic D/V shown in double-line rectangle is finally obtained. The membership function (/ Yi/w of D/V represents the optimum degree of aseismic structural characteristics. In these case studies, fuzzy inputs and fuzzy states are calculated by solving fuzzy relational equations [14][11].

113 ;V/R r

200 100 75 (Vear)

;f\|

25 20 15 10 5(Kine) !

v ^ l 30 20 10 5

200 0.3 I 0.3 0 0 100 0 0.3 1 0.3 0 15 0 0 0.3 1 0.3

/ i j ' O . ! 1 0.2

I ^'^D/R

D/V

; (Year)

1 25 20 0.3 0 15 0 10 0 5 (Kine)

"MA

i

Z(cii)

U. 10.3 0.3

4 l

0 0.25 0.5 0.75 1.0

• 30 0.3 1 0.3 0 0 : 20 0 0.3 1 0.3 0 : r=200 10 0 0 0.3 1 0.3 :(Year) 5 0 0 0 0.3 1 I 2 0 0 0 0 1

0.3 0 0 0 1 0.3 0 0 0.3 1 0.3 0 0 0.3 1 0.3 0 0 0.3 1

..'M

:

'"'

;

dS

"5/0/200 0 0.25 0.5 0.75 1.0

; : ;r=100 i(year) ;

90 20 10 6 2

i

<'">

;

(Ti

: ; : r=75 iCYear) ;

30 1 0 0 0 0 1 0.3 0 0 0 20 10 0.3 1 0.3 0 0 5 0 0.3 1 0.3 0 2 0 0 0.3 1 0.3

1 0.3 0 0 0

0.3 1 0.3 0 0

0 0 0.3 0 1 0.3 0.3 1 0 0.8

0 0 0 0.3 • 1 ;

"S/B/Iooi 0 0.25 0.5 0.75 1.0 ! ; i ; ; ;

1 0.3 0.3

i <"' "sml ., Fig.30 Calling and Reasoning Flows in the case of Calling Structural Safcty(S) ^

D/V

; v/R r

25 20 15 10 mne) ;

200 100 75 (Year)

/ij'o.Z

; (Year)

; Input InDut

;L

L

AT

tm

: :

Call /y

flte

f V/ ' > /R<

' D/ 1

&

D/R 3D 20

25 1 1 20 1 1 15 0.3 0.5 10 1 1

0.3 1 0.3 0 0 100 0 0.3 1 0.3 0 ?i 0 0 0.3 1 0.3

1 0.2

: SI

v:ij

5 1 1 (Kine)

0 5 2(cm) 1 1 0.5 1 1

1 1 0.3 1 1 "D/V

:

3D ri

Input

Input

AT

AT

S/

:s

Result of

; Knotledge

Beasoning

;s • s I 0 0.25 0.5 0.75 1.0 i;;

10.3 0.5

i ;

30 20 10 5 z <"> d ^

1 0.5 0.3

0 0.25 0.5 0.75 l.i 0.3 I 0 0.3 0 0 0 0 0 0

0.3 0 1 0.3 0.3 1 0 0.3 0 0

;

30

;

20

;r=75 10 •(Year) 5 : 1 ;

0 0 0.3 1 1

"s/D/200 0 0.25 0.5 0.75 1.0

1 0.3 0 0 : 20 0.3 1 0.3 0 ;r=100 10 0 0.3 1 0.3 ;(Year) 5 0 0 0.3 1 ; 2 0 0 0 0.3 i (ci) (Txl

IJRrr^

: Input

d\| ; ; ;r=200 ;(Year) :

0 0 0 0.3 1

"S/D/100 0 0.25 0.5 0.75 1.0

1 1 0.3 0 0

0 0 0 0 0.3 0 0 0 1 0.3 0 0 0.3 1 0.3 0 0 0.3 1 0.3

(cii)

Fig.31 Calling and Reasoning Flows in the case of Calling Aseismic Structural Characteristics (D/V)

.."jMi.l

114

3-4-2. Inference Tree Method [9] In this case study, a standard type of reinforced concrete (R/C) transfer substations of Kansai Electric Power Co. Ltd. is employed for aseismic structural planning. This building is assumed to be located at Kobe City in Japan. Aseismic optimization and evaluation are performed in the X-longitudinal direction. Fig.32 shows a fuzzy network composed of undirected fuzzy systems for seismic planning of R/C building. • and [Q] mean variables and states, respectively. The variables are defined as shown in Table 7. The states arc already given from seismic and structural points of view [9]. Aseismic optimization is performed with respect to optimal number of shear wall units. If N-WALL in Fig.32 is nominated for the final output, then the fuzzy network is transformed into a fuzzy hierarchy such as shown in Fig.33. Considering the properties of the R/C structure and using the inputs described with fiizzy sets shown in Fig.34, calculated variables in structural optimization are obtained as shown in Fig.35. Fig.36 shows the region of optimal numbers of shear wall units illustrated with membership functions given by maximizing decision [1], which means the optimal number is about 9-10.

MS-I fOTfj^M-CEQ d= ^^IMWIMI

STATHEfflil

IM-WAG

m

MB^Mm

mm

: STATE

MM-MM-^M Fig.32 Fuzzy Network for Structural Planning

Table 7 Variables in Fuzzy Network for Structural Planning N-COLM

Number of Columns

A-WALL

:Sectional Area of Shear Walls

D-COLM

Depth of Column Sections

ACC

Acceleration Amplitude of Ground Motions

HHGHT

Height of Columns

,N-WAli

iNecessary Number of Shear Wall Units

WEIGHT

Seismic Weight of BuOdings

,W-DESIGN

:Design Length of Opening of Shear Walls

l-WALL

Thickness of Shear Walls

,W-NECE

iNecessary Length of Opening of Shear Walls

L-WALL

Span Length ot Shear Walls

.EOONOMY

:Satisfactory Degree of Economy

N-ALL

Total Number of Shear Wall Units

.SAFETY

:Satisfactory Degree of Safety

.FUNCTION

[Satisfactory Degree of Function

.PERIOD

:Retum Period

.DAMAGE

iDamage Factor

,1

:lmpOTlance Factor

Q-R-COLM Force-Deformation Relation ot Columns A-COLM

Sectional Area of Columns

Q-R-WAIX Force-Deformation Relation of Shear Walls

115

Aseismic evaluation is performed regarding economy, safety and function. If ECONOMY, SAFETY and FUNCTION are nominated for the final outputs, the fuzzy network in Fig.32 is transformed into a fuzzy hierarchy such as shown in Fig.37. Using the inputs described with fuzzy sets shown in Fig.38, Fig.39 shows the final evaluations of economy, safety and function illustrated with fuzzy sets given by maximizing decision, which means they are good conditions except safety.

iKMETMTlFMffllPERTillEM

•

INPUTS

Fig.33 Transformed Fuzzy Hierarchy for Structural Optimization

Ql—'—i—•—^WEIGHT nl—.—i—.—5.N-COLM n 2200 (t) " 22 " p. p. fl ID 10 1

„

->—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<(i-l)

(3)

where the learning rate e controls the descent, 0 < a < 1 is the damping coefficient or momentum and t denotes the number of the iteration currently in progress.

B.

Input vector representation

An n-dimensional pattern F, = [-fii, -fizi •••> Ein] can be represented as a 3n-dimensional vector

iF<}]

(4)

where the /x values indicate the membership functions to the corresponding linguistic TT-set [18] for eeich feature axis.

131

When Fj is numerical we use the ff-fuzzy sets [66] (in the one-dimensional form), with range [0,1], given as • 2 ( l - M ) ' , 7r(F,-;c,A) = • 1 _ 2 ( M ) \ 0,

for|<|f,-c|
(5)

where A > 0 is the radius of the 7r-function with c as the central point. When the input feature is linguistic, its membership values for the 7r-sets low, medium and high are quantified as fO.95 0.6 0.021 fO.7 0.95 0.71 \ L' M 'H i r0.02 0.6 0.951

C.

Output vector representation

In real-life problems, the data are generally ill-defined with overlapping or fuzzy class boundaries such that each pattern used in training may possess finite belonglngness to more than one class. To model-such data we clamp the desired membership values, lying in the range [0,1], at the output nodes during training. For an i-class problem domain, the membership of the :"' pattern to class C^, lying in the range [0,1], is defined as

where 2,^ is the weighted distance between the t"" pattern and the mean of the fc"* class and the positive constants Fj and F^ are the denominational and exponential fuzzy generators controlling the amount of fuzziness in this class-membership set. Then for the x"' input pattern we define the desired output of the j ' " * output node as dy-My(^.) (7) During testing, the output of the j " " output neuron denotes the inferred membership value of a test pattern to the y"" class.

D.

T-Norm and T-Conorm

We consider a special case of the conjugate pair of t-norm T and t-conorm S represented by the And and Or nodes at the hidden and output layers respectively. The product and probabilistic sum operators are given as T{a,b)=ab

132

S{a,b).= a + b-ab

(8)

Let us consider the three-layered model as shown in Fig. 1 with H = 2. The output j/j of the j " * neuron in the first layer is defined as

y]= .r[5(y°,OJ

(9)

Here the input vector is in the 3n-dimensional space of eqn. (4) such that t/° is the output of the »"" neuron in the input layer. The T operation is performed over all S operation outputs corresponding to the 3n neurons in the input layer. Analogously, the output yl in the second layer is given as

yl= ,s[T(y},wl,)]

(10)

where the 5 operation is performed over all j T operation outputs corresponding to the neurons in the first layer.

E.

The backpropagation algorithm based on logical operations

The LMS error E of eqn. (2) is minimized by the gradient-descent technique of eqn. (3) with the restriction that 0 < ly^,- < 1 for 0 < h < 2. The error derivative ^ ^ is computed as dE _ dE dyj dwji

(11)

dyj dwji

For the output layer [h = H) we substitute dE

Byr'^-'^

(12)

In case of the hidden layer we use

dVj

^dyt

dyj

Y

^J/J

where neurons j and k lie in layers h and /i + 1 respectively. In order to evaluate the derivative ^ ^ of eqn. ( l l ) , for /i > 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^<^^ ^^^^ '^i = (ui — l) * 3 + 1. The response from an unobtainable input variable can allow the neuron activations in the following layers to become non ambiguous with noinf^ = 0 such that an output decision may finally be inferred.

D.

Justification

In this phase the user can ask the system why it inferred a particular conclusion. The system answers with an // - Tlien rule, applicable to the case at hand, in terms of the salient input features. It is to be noted that these If - Then rules are not represented explicitly in the encoded knowledge base; they are generated by the inferencing system from the connection weights as and when needed for explanations. As the model has already inferred a conclusion (in this stage), we take a subset of the currently known input information to justify this decision. Let the user ask for justification about a conclusion regarding class j . We choose neuron t from the hidden layer fT — 1 if w^-~^ > 0, i.e., if it has a positive impact on the conclusion at output neuron j . Let a set of nin-i neurons be so selected. For the remaining layers we obtain the maximum weighted paths through these neurons down to the input layer. Let the set of the selected mo input neurons be given by la^, a ° , . . . , a^^ | and their corresponding path weights to neuron j in layer H be denoted as

Iwet^OjWet^o,...,

wet^^o^ \. We arrange these neurons in the decreasing order of their net impacts, where we define the net impax;t for neuron : as net impacti — y,° * wetio Then we generate clauses for an If-Then rule from this ordered list until J2wetio>2j2^etio

(27)

where i, indicates the input neurons selected for the clauses and t„ denotes the input neurons remaining from the set j a j , a°,..., a^\. This heuristic allows selection of

137

those currently active input neurons (current evidence of test patterns) contributing the most to the final conclusion (ajnong those lying along the maximum weighted paths of the embedded knowledge base to the output node j) as the clauses of the antecedent part of a rule to justify the current inference. Let u,, be the input feature corresponding to a neuron t',, in the input layer [h — 0), selected for clause generation. The antecedent of the rule is given in linguistic form with the linguistic property being determined cis [ low prop — < medium [ high

if t „ - 3fu,. - 1) = 1 if t„ — 3(u,, — l) = 2 otherwise

(28)

A linguistic hedge very, more or less or not may also be attached to the linguistic property obtained for the antecedent part. We use the mean square distance
E.

Implementation

and results

The three-layered logical neural net was used on vowel data consisting of a set of 871 Indian Telugu vowel sounds uttered in a Consonant-Vowel-Consonant context by three male speakers. Fig. 3 shows a 2D projection of the 3D feature space of the six vowel classes (5, a, »', u, e, o) in the Fi — F^ formant frequency plane. We selected Fj = 5, F, = 1 in eqn. (6) and kn = 10, 6 = 0.0001 in eqn. (19) after several experiments. Table 1 demonstrates the rule generating ability of the trained logical neural model (with one hidden layer having 20 nodes), that functions as a knowledge base for the vowel recognition problem, using a sample set of test input feature combinations. It was trained using 10% samples from each representative class.

138 900 SIZE

n

800

FREH^UENCY OF OCCURENCES

1 3 S !0 15

— 2 — 5 — 9 —U MiD ABOVE.

70 C BOO ' 50Cc

iOG

30C 2CC

600

SCO

uQQ

!500

1300 Fj in Hz

2100

2
2700

F i g u r e 3 : Vowel diagram in the F^rF^ plane.

VI.

Conclusion and Discussion

Logical operators like And and Or are seen to ease the computational burden of the neural model to some extent, thereby enhancing its speed (This will enable simpler hardware implementation for pattern recognition problenis). Besides, the fuzzy logical model is also able to generate more appropriate rules (expressed as disjunction of conjunctive clauses) from the knowledge base embedded among the connection weights as compared to those obtained from its fuzzy counterpart mcorporating no logical operators [68]. This is due to the b u i l t - m W - O r structure of the network architecture under consideration. The model is likely to be suitable in data-rich environments for designing classificatton-type expert systems, i h e service of human experts can be utilised in verifying the justificatory rules generated by the model, instead of designing the rules for the knowledge base (as is done m case of the traditional models). It is to be noted that the input feature values of a test pattern influence the generation of the rule from the trained set of connection weights. This helps extracting rules relevant to that region of the feature space which is local to the area pointed to by the feature values of the test pattern.

139

Table 1 : Rule Generation for Vowel data with Fuzzy Logical Model benai No.

ri

i n p u t features J-'a

i-'s xinobt

1

500

soo

3

700

1000

2600

3

700

nnobt

musing

4

400

800

unobt

5

400

unobt

unobt

700

1300

unobt

600

1200

unobt

450

2400

unobt

9

900

1400

unobt

10

1600

missing

11

bctwien 500 d: 600 greaier t/um 650

high

missing

12

o6ffui350

uno6t

unobt

6 7 8

Justification / K u l e g e n e r a t i o n ij clause iTien c o n c l u s i o n ^'i is very low or very likely class u F^ is very ^o^ Fz is very medium and Fi is very medium or F i is Mol >uyfc very Ukely class a or ^2 is very tow likely class a, b u t F j is very medium nof un/ubeiy class e or Mol ki^ very likely class o, b u t Fi is very low and not unlikely class u Fa is very low hfol likely classes e or i, b u t F\ is very low not unii%e/i/ class o Fa is very medium and F i is very rr^edium very likely class a or F\ is Mol hi^ Fi is very medium and F a is pery medium Mol liiily classes a or o or Fa is low F a is very feV^ a n d likely class i, b u t F\ is A/(3i /ou; not unlikely class e or Fi is very medium Fi is pffry mtfJj'um o r very likely class a F j is wry Ai'yA likely class e, b u t Fa is very rnedium or not unlikely class o F i is very milium or Mo/ low Fa is /ujjA a n d F i is A/tjf AiyA t>try likely class e or F i is eery miiii'um vfry ibt«^y class t , b u t F i is very law not urJikely class e

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14) B. Kosko, Neural Networks and Fuzzy Systems. New Jersey; Prentice Hall, 1991. 15] Proc. of Second International Conference on Fuzzy Logic and Neural (UZUKA92), (lizuka, Fukuoka, Japan), July 1992.

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CHAPTER 7

FUZZY REASONING THROUGH A GENERAL NEURAL NETWORK MODEL S.G. Tzafestas(*), G. B. Stamou(*) and (*) (**)

K. WatanabeC*)

Intelligent Robotics and Control Unit National Technical University of Athens Zografou Campus, Zografou 15773, Athens, Greece Department of Mechanical Engineering Saga University Honjomachi-1, Saga 840, Japan

1. Introduction Neural networks or connectionist models are designed to emulate human performance at high computation rates through appropriate massively parallel interconnections of simple neurons. On the other hand, fuzzy systems are capable to model imcertain situations that occur in real life environments. One way of implementing fuzzy reasoning systems is through expert system architectures, the alternative being through neural network structures. Neural systems and fuzzy systems have essential differences in the way they function. Neural systems are based on pattern recognition supervised or unsupervised learning techniques , whereas fiizzy systems are based on fuzzy set theory and fuzzy inference mechanisms. However neural and fuzzy systems have in common two important characteristics, namely: i) they both possess a high degree of parallelism, and ii) they both belong to the class of the model free estimators. Fig. 1 shows the position of fiizzy and neural systems according to the knowledge framework (symbolic, nimieric) and knowledge type (structured, unstructured) taxonomies. It is clear that the combination of fiizzy and neural systems is possible and indeed there are currently available several forms of fiizzy-neural systems [1-5]. Our purpose here is to present and study a convenient neural network which implements a kind of fuzzy inference. The system is actually based on the Keller-Yager-Tahani (K-Y-T) neuro fuzzy model and uses Hamacher's intersection function l^{a,b) or Sugeno's complement fijnction C;i(a), 145 S.G. Tzafestas and A.N. Venetsanopoulos (eds.) Fuzzy Reasoning in Information, Decision and Control Systems, 145-161. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.

146

We start with a brief description of the system operation and then establish its key features in the form of four theorems which are fully proven. The capabilities of the system are illustrated with the aid of a rich set of simulation results derived for the case of trapezoidal fiizzy sets. These results are shown to be better than the ones obtained with the original KY-T systems.

Knowledge Structured Symbolic

Unstructured

AI Expert Systems

Framework Numerical

Fuzzy Systems

Neural Systems

Fig. J: Taxonomies ofAI Expert, Fuzzy and Neural systems. Each neural net is known to consist of a set of elementary units with n synapses and weights wjj which are connected to a node f described by a strongly nonlinear operation. These units are refered to as "neurons" (Fig.2).

OUTPUT-

x

Wii

}

INPUTS

Fiii.2: Basic structure of a neuron.

The purpose of any learning procedure is to update weights wjj such that for each input for the neural network, the output to be the desired one. Actually, a plethora of neural network models and learning algorithms have been proposed in the technical literature (see e.g. [611 ]). Of course there exist many situations where the weights are selected right from the beginning by the designer without the application of any learning algorithms. The system presented here belongs to this category.

147

The fijzzy systetns were intoduced by Zadeh in 1965 [12] and have shown a significant development due to their ability for solving complex reasoning problems. They are based on fuzzy logic and fuzzy sets. The fundamental operations of fuzzy sets are: fuzzy complement, fuzzy union and fijzzy intersection [13]. With the aid of these operations, the extension of logical tautologies becomes possible. A particular tautology which is directly useful in fuzzy controUers is the modxis ponens (MP) rule: [{A

-^B)AA)^B

which indicates that in a cause-effect relationship, when the cause is satisfied the consequence is obtained. The question is "what happens when A and B are fuzzy sets?". In this case the above MP rule is extended to the following generalized modus ponens (GMP) rule: {{A ->B)A

A') -> B'

which shows that when the fiizzy cause is ahnost satisfied we nearly get the consequence. One way to implement the GMP rule is through the use of Zadeh's max-min composition operator [12,18]. A fuzzy controller consists of the following parts (Fig.3) [6,14,18]: - fuzzification - fuzzy inference mechanism - defiizzification

-0--

Process

n

Defuzzificatlon

J

Fuzzy Inference L-' Mechanism J

n

Fuzzlftcatton

Fuzzy Knowledge Base

Fuzzy Controller Fig.3: General structure of a fiizzy coBtroller. The fuzzy inference mechanism is implemented with the aid of a fiizzv knowledge base, i.e. a knowledge base that contains fuzzy rules (MIMO: Multi-fnput/Multi-Output rules), ft is

148

shown in [16] that these rules can be transformed to MISO (Multi-Input/Single-Output) rules. If the knowledge base contains many such rules, then one can assume that the inference is done in the union of them. It can be shown [16] that it is sufficient to apply the input to every rule and get as result the union of the outputs. Thus the design of rules of the type; IF Xi IS Aj AND X2 IS A2 AND ... X„ IS A„ THEN Y IS B

(I)

plays an important role in the design of a fuzzy controller. The inference is performed by using the GMP rule.

2. Fuzzy inference via neural nets From the discussion of the previous section it is clear that there are some basic similarities between neural and fuzzy systems. Our purpose in the present section is to try to use a neural net for drawing fuzzy conclusions. To this end we shall extend and improve the solution proposed by Keller, Yager and Taham [15]. This sohjtion has the drawback that, although it possesses a correct behaviour, it is away from the desired solution, especially when the fact (event) occurs. Thus an improvement will be proposed here together with a discussion on the application domain of the network. Actually, one of the main consequences of the use of fuzzy reasoning in a rule based (IF-THEN) system is the dramatic reduction of the number of rules which leads, among others, to faster conclusions. Some of the available solutions for the organization of fuzzy controllers are those proposed by Procy and Mamdani [20], Braae and Rutherford [21] and Macvicar-Whelan [22], where the fiizzy matrix concept is utilized for the representation of the fuzzy control relationships of concern. These solutions give very good results but possess the following drawbacks : i) They require a large memory space for the storage of the elements of the fuzzy matrices (but they gain in comutation time). ii) The fuzzification unit must produce results that are compatible with the representation of the variables in the fuzzy matrices. For example if the input is quntized as "LOW, MORL LOW, MEDIUM, MORL HIGH, HIGH", the output of the fuzzifier must give one of the values "LOW, MORL LOW, MEDIUM, MORL HIGH, HIGH". This drawback is weakened when the quntization step is reduced, but then again the required memory increases.

149

In our system, no ftizzy matrix is used, but instead a sequence of fuzzy operations (fuzzy algorithm). In this way the extra memory for the fuzzy matrix is no more required, and resulting increase of the computational time is faced by employing a neural network structure that has a high degree of parallelism. An additional benefit of the method presented here is the "continuity" in the sense that the system is able to respond without any particular care, to any input value that may not belong to the value set {LOW, MORL LOW, MEDIUM, MORL HIGH, HIGH}. The proposed neuro-fuzzy network must satisfy the following requirements: a. The fiizzy sets must be given in sampled form, since every neural network has a finite number of inputs, i.e. Aj and B in the rule (1) must be of the form: A = {^/l.•••.3*1,}. " = mi,...,m„

for / = l,...,n

B = {/>,....,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(
b; = c{nu,,s)}

150

where c: involutivc fuzzy complement with the property c(c[a)) = a, Va, u; fuzzy union with the property u(a,0) = a and u{a,l) = 1, /: fuzzy intersection with the property /(a,0) = 0 and /(a,l) = a, satisfies the above conditions a. and b.

^1

^m,

^1

Kn\,

Fig.4: Structure of the proposed neural network (assuming that e,=l, V/ = l,2 k)

Theorem 1 UA;

- A, for all / = 1,2,.... n, then the result is B.

Proof From Aj = A,' it follows that a^ = a]j for all / = 1,2,..., n and so for all v,j in the domain of discourse it is true that tiW^) = a^ = 1 or 0 and //(c(i/,^)) = 0 or 1, respectively, where

151

p{x) is the membership function of concern. That is V/ = l,2,...,n the following relation holds: d^ = u{na,j,c{aij))} = 0 j

since /(a,y,c(a,y))} = /(l.O) = 0 for all i andj of interest. Thus s = c{u(rfk)} = c(0) = l k

and so b- = c{i(Uj,s)} = c(u,) = c(c(fa,)) = 6, for / = 1,2

n. This means that the result

is B as required, q.e.d. Theorem 2 If for some i from 1 to n the condition A^ nAI^O holds, then the neural network produces the result "y is UNKNOWN" (which means that £>/ = 1 V/ = 1,2,..., n). Proof Since fJ cWjj) = f^A'^^ij) = 1 ^or some vij in the domain of discourse and /(a,y,M',y) = 1 for this Vfj,

du=u{na;j,wy)}

=\

Thus s = c{u[d,,)} = 0 and so b; = c{/{0,U;)} = c{0) = 1, V/ = l,2,...,n. q.e.d. k

Theorem 3 If Al Q Af (i.e. Al is more specific than A/- ft A. ^ PA) for some i from 1 to n, then the result is B.

Proof From Al c A, for some i, it follows that ^ nA^

= 0 . Hence d/, = 0 which gives the

result "Y is B". q.e.b. This network is used to solve problems similar to the following example [15]. Consider the rule IF a person is OLD THEN the person in UNHEALTHY

152 If some time we learn that the person is VERY OLD, then this information is sufficient to say that the person in UNHEALTHY. To confirm this fact the following theorem will be shown.

Theorem 4 Consider two inputs Aj and /I" / = 1,2

n of the neural network and suppose that

A " £ A' £ A ^°^ all / = l,2,...,n. Then the corresponding results B' and B" will satisfy the condition B c B" c B' Proof From the relation /V'e A G A . V/ = 1,2,...,n, it follows that 0 < /i^.(-) < p^.(-) < fiA, (-). V/ = 1,2,...,n for any point in the domain of discourse. Thus dH = u{/K.iv,^)}
and so if c/j^ < dji then u((^it) k

J U(<^A^) k

which implies that

c{uK)}
k

or s' < s" Thus we have 6/ > 6," i.e. B c B" ^ B' because 6/,6;'> 6,, V/ = 1,2

n where

6/ = c{/(y,,:j)} = c{/(c(6,-),s)} > c{c(6,)} = ft,

Remark For any input Aj it will be true that B c B'. It is therefore clear that for any triple of functions (c, u, i) the network will be working for examples of Che "unhealthy old person" type mentioned above. The problem now is to select the functions u(-) and i(-,-) that satisfy the above properties. Actually, many functions are appropriate for this purpose [13]. A first selection is the one of Keller, Yager and Tahani [15]. Looking carefully at the neural net of Fig.4, one concludes that the dominant role is played by the second player where the d^'s are

153

computed. The core of this operation is the comparison of the input with the prototype that is represented by the weights Wjj, i.e. the event i of the GMP rule. To achieve this, the complement of this event is stored in the Wy's and the maximum element of each intersection with the input is obtained. The function of this intersection must be as "hard" as possible, so as to give a value as near to zero as possible if there exists a relation of the input and the event. Otherwise it must give the value 1, which means that a decision cannot be obtained and is thus left to another GMP rule. Most of the fuzzy function classes for the intersection, give the possibility for such a "hard" intersection under a suitable choise of the values of their parameters. The final selection of the intersection fimction among them, is made on the bases of the validity of the conditions mentioned above and used in the proofs of theorems I through 4, and the computation time required to execute the mathematical operations that support it. In our system the following neuron model is selected in order to possess all desired features:

d^ -

max.[iH(3ij,Wij)) J

s = 1- max {dif} k

u, =

l-b,

bj = 1 - s • Uj

where //y(-) is Hamacher's ftmction [13]:

y + {l- y)[a + b- ab) which satisfies the conditions //y(a, 1) = a and if^(a,0) = 0. As it will be seen in the next section, this model works correctly for relatively large values of y

(Y=100.

200,...).

However, there exist rales of other types that are also compatible with the GMP rale. To see this we give the following example. Suppose that two taps feed water to a vessel. The temperature of the water flowing from the first tap is known but not controlled, whereas the temerature of the water in the second tap is controlled with a switch that has the following indications: VERY VERY WARM,

VERY

WARM,

WARM,

NEARLY

WARM,

A

LITTLE

WARM,

154

MODERATELY WARM. A LITTLE COOL, NEARLY COOL, COOL, VERY COOL, VERY VERY COOL. We want to keep the water in the vessel MODERATELY WARM. The logic of the previous model is as follows. If the water of tap 1 is A LITTLE WARM or NEARLY WARM or VERY WARM or VERY VERY WARM then open the COOL water in tap 2. In this way, we will get the correct result with a delay. It would be better to work in the following way. If the water in tap 1 is WARM then open in tap 2 the COOL water, if it is VERY WARM then open the VERY COOL water and if it is VERY VERY WARM then open in tap 2 the VERY VERY COOL water, and so on. In this way we will get the desired MODERATELY WARM water very fast. However, the proposed inference model cannot produce this logic due to Theorem 4. Of course if in the computation of the ui's one uses a different function for the complement operation than that used for the bi"s, then it could be possible to obtain 6,' < b,. But in this case, the proofs of Theorems 1, 2 and 3 should be modified. One must therefore be very carefull in using this model in each particular case. In the light of the above, we propose here the following modification of the inference network:

dff = Ti\av.[a'fj,w^) s = Ci(max(
u, = C2(bi) bi = c-^(sui) where Cj and C2 are complementation functions of the Sugeno type [13]: C/i(a) =

— , ?i e ( - l , Q o )

\ + Aa

In practice one must select suitable values for \ and /Ig - This choise can be made through simulation as described below.

155

Table 1: The weaning of linguistic terms defined on the domain [1,11] and sampled at integer points.

Label LOW VERY LOW MORLLOW NOT LOW MEDIUM MORL MEDIUM NOT MEDIUM HIGH VERY HIGH MORL HIGH NOT HIGH UNKNOWN

Values of the membership function 1.00 1.00 1.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 1.00 1.00

0.67 0.45 0.82 0.33 0.00 0.00 1.00 0.00 0.00 0.00 1.00 1.00

0.33 0.11 0.57 0.67 0.25 0.50 0.75 0.00 0.00 0.00 1.00 1.00

0.00 0.00 0.00 1.00 0.50 0.71 0.50 0.00 0.00 0.00 1.00 1.00

0.00 0.00 0.00 1.00 0.75 0.87 0.25 0.00 0.00 0.00 1.00 1.00

0.00 0.00 0.00 1.00 1.00 1.00 0.00 0.00 0.00 0.00 1.00 1.00

0.00 0.00 0.00 1.00 0.75 0.87 0.25 0.20 0.04 0.45 0.80 1.00

0.00 0.00 0.00 1.00 0.50 0.71 0.50 0.40 0.16 0.63 0.60 1.00

0.00 0.00 0.00 1.00 0.25 0.50 0.75 0.60 0.36 0.77 0.40 1.00

0.00 0.00 0.00 1.00 0.00 0.00 1.00 0.80 0.64 0.89 0.20 1.00

0.00 0.00 0.00 1.00 0.00 0.00 1.00 1.00 1.00 1.00 0.00 1.00

3. Simulation example The proposed neural networks were simulated for trapezoidal fuzzy sets [17] according to the rule: IF X IS LOW THEN Y IS HIGH and the rule: IF X, IS LOW AND Xj IS LOW THEN Y IS HIGH. The results obtained are shown in Tables 2 through 8. We observe that they approach the desired result, the first without deviation and the second with some tolerable deviation. Table 2: The results oftite simulation of the first model using Hamacber's function (y=50)

Label LOW MORLLOW VERY LOW MEDIUM NOT LOW

Va ues of the membership function 0.02 0.05 0.01 1.00 1.00

0.02 0.05 0.01 1.00 l.QO

0.02 0.05 0.01 1.00 1.00

0.02 0.05 0.01 1.00 1.00

0.02 0.05 0.01 1.00 1.00

0.02 0.05 0.01 1.00 1.00

0.21 0.24 0.21 1.00 1.00

0.41 0.43 0.40 1.00 1.00

| 0.61 0.62 0.60 1.00 1.00

0.80 0.81 0.80 1.00 1.00

1.001 1.001 1.001 1.001

1.001

156

Table 3: The results of the simulation of the first model using Hamacher's function (Y=100)

Label LOW MORL LOW VERY LOW MEDIUM NOT LOW

1

Values of the membership function 0.01 0.03 0.00 1.00 1.00

0.01 0.03 0.00 1.00 1.00

0.01 0.03 0.00 1.00 1.00

0.01 0.03 0.00 1.00 1.00

0.01 0.03 0.00 1.00 1.00

0.01 0.03 0.00 1.00 1.00

0.21 0.22 0.20 1.00 1.00

0.41 0.42 0.40 1.00 1.00

0.60 0.61 0.60 1.00 1.00

0.80 0.81 0.80 t.OO 1.00

tool i.ool i.ool I.ool I.ool

Table 4: The results of the simulatioa of die first model using Hamacher's function (r=200) Label LOW MORL LOW VERY LOW MEDIUM NOT LOW

Va ues of the membership function 0.00 0.01 0.01 1.00 1.00

0.00 0.01 0.01 t.OO 1.00

0.00 0.01 0.01 1.00 1.00

0.00 0.01 0.01 l.OO 1.00

0.00 0.01 0.01 l.OO l.OO

0.00 0.01 0.01 l.OO 1.00

0.20 0.21 0.21 l.OO 1.00

0.40 0.41 0.40 1.00 l.OO

0.60 0.61 0.60 1.00 l.OO

0.80 0.80 0.80 1.00 l.OO

| I.ool I.ool 1.001

I.ool I.ool

's function "able 5: The results of the simuladon of die second model using Sugeno Values of the membership function | Label 0.05 0.05 0.05 0.05 0.05 1 0.05 0.24 0.42 0.61 0.80 I.ool LOW 0.84 I.ool 0.09 0.09 0.09 0.09 0.30 0.49 0.67 0.09 0.09 MORL LOW 0.54 0.02 0.02 0.02 0.02 0.02 0.02 0.18 0.35 0.76 I.ool VERY LOW MEDIUM 1.00 1.00 l.OO 1.00 t.OO 1.00 t.OO 1.00 1.00 too I.ool NOT LOW t.OO 1.00 1.00 1.00 1 1.00 t.OO 1.00 l.OO 1.00 l.OO tool Table 6: The results of the simulation of the first model using Hamacher's function (y=50. two inputs/one output) Values of the membership function Label LOW. LOW MORL LOW, LOW MORL LOW, MORL LOW VERY LOW, LOW VERY LOW. VERY LOW VERY LOW, MORL LOW VERY LOW, MEDIUM

0.02

0.02

0.02

0.02

0.02

0.02

0.21

0.41

0.61

0.80

1.00

0.05

0.05

0.05

0.05

0.05

0.05

0.24

0.43

0.62

0.81

1.00

0.05

0.05

0.05

0.05

0.05

0.05

0.24

0.43

0.62

0.81

1.00

0.02

0.02

0.02

0.02

0.02

0.02

0.21

0.41

0.61

0.80

1.00

0.01

0.01

0.01

0.01

0.01

0.01

0.21

0.40

0.60

0.80

1.00

0.05

0.05

0.05

0.05

0.05

0.05

0.24

0.43

0.62

0.81

1.00

1.00

l.OO

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

157

Table 7: The results of the simulation of the first model using Hamacher's function (y=IOO. two inputs/one output)

Label LOW. LOW MORL LOW. LOW MORL LOW, MORL LOW VERY LOW. LOW VERY LOW, VERY LOW VERY LOW, MORL LOW VERY LOW. MEDIUM

Values of the membership function 0.01

0.01

0.01

O.OI

0.01

0.01

0.21

0.41

0.60

0.80

1.00

0.03

0.03

0.03

0.03

0.03

0.03

0.22

0.42

0.61

0.81

1.00

0.03

0.03

0.03

0.03

0.03

0.03

0.22

0.42

0.61

0.81

1.00

0.01

0.01

0.01

0.01

0.01

0.01

0.21

0.41

0.60

0.80

1.00

0.00

0.00

0.00

0.00

0.00

0.00

0.20

0.40

0.60

0.80

1.00

0.03

0.03

0.03

0.03

0.03

0.03

0.22

0.42

0.61

0.81

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1,00

1.00

1.00

Table 8: The results of the simulation of the first model using Hamacher's function (y=200, two inputs/one output)

Label LOW, LOW MORL LOW, LOW MORL LOW, MORL LOW VERY LOW, LOW VERY LOW. VERY LOW VERY LOW, MORL LOW VERY LOW, MEDIUM

Va ues of the members!lip function 0.00

0.00

0.00

0.00

0.00

0.00

0.20

0.40

0.60

0.80

1.00

0.01

0.01

0.01

0.01

0.01

0.01

0.21

0.41

0.61

0.80

1.00

0.01

0.01

0.01

0.01

0.01

0.01

0.21

0.41

0.61

0.80

1.00

0.00

0.00

0.00

0.00

0.00

0.00

0.20

0.40

0.60

0.80

1.00

0.00

0.00

0.00

0.00

0.00

0.00

0.20

0.40

0.60

0.80

1.00

0.01

0.01

0.01

0.01

0.01

0.01

0.21

0.41

0.61

0.80

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

In the first case (Tables 2-4) we present the results of the network for the Hamacher ftmction with y=50, 100 and 200, respectively (the plots are omitted since they almost coincide with the desired results). In the second case (Table 5) we use the complement function of Sugeno with /Ij = 10 and /Ig = 6 and give the corresponding graphical representations. In the third case (Tables 6-8), we present the results of the two inputs/one output network for the Hamacher function with Y=50, 100 and 200 respectively.

158 1 0,9 0J8 0,7 0,6 0,5 0.4 0,3 0,2 0,1 0

Output of low (KY-T) • - — • Output of very tow (K-Y-T) • Output of morl low (K-Y-T) Higfi (K-Y-T) Output of morl low (1st model V=200) 1

I

2

3

S

6

7

8

9

10

U

Fig.5: Comparison between the first model (y=200) and the K-Y-T model (the outputs of hw and very low of the first model are almost identical to High)

4. Conclusions The neural network proposed here has the same structure as the K-Y-T model but it is generalized such that one can use any meaningful fitzzy intersection, union or complement ftmction. This is reflected in the more generality of Theorems 1 through 4. Here we selected and used the Hamacher intersection function which leads to moie accurate inferences than those obtained by the K-Y-T network. Also we used the Sugeno complement function which gave us an other improvement A further extension would be obtained if one uses rules of the type: "IF water is WARM THEN open COOL tap", "IF water is VERY WARM THEN open VERY COOL tap", and so on (as mentioned at the end of section 2). The authors are currently working to design a system that can accomodate such rules and also in the area of tuning PID and otilier controllers with die aid of the present inference system in place of the so called "fuzzy matrices" [19]. Hopefully the use of the present system will give equivalent tuning results as in [19] but more rapidly and with increased robustness. Work is also in progress for enriching the structure with a learning capability. Acknowledgement This chapter is based on a paper presented at the AIENG 93: Applications of Artificial Intelligence in Engineering (Toulouse, June-July 1993).

159 1 0.9 0.8 0.7 + 0.6 03 0,4 • 0.3.. 0.2-

y

.' / • '

/

Mori high Output of morl bw (2nd mod«Q

oa^.

. J-

-i-

0

H

-*8

9

10 11

0.903 0.7 0,6 0,5 0,4

Very high Output of very tow (2nd modeO

0^1 0,2

oa 0 - - t - T - r - I — I - if • ' i 5 6 7 2 3

8

-+-

-1-

9

10

-H

U

1 0,9 0,8 0,70,6 0^.. 0,40,3

High • — — Output of low (2nd model)

o;? 0

-i

1

1

2

3

~r'

»-

4

5

6

I 7

-i

8

H

9

10

H

F>fl6: Comparison between the output of tt)e second model and the desired one.

160

References [11 T. Yamakawa, "PatteaTi Recognition Hardware System EmpJoying a Fuzzy Neuron', Proceedings of the latematioaal Conference on Fuzzy Logic and Neural Networks (Izuka, Japan, July 20-24, 1990) pp. 943-948. [2] H. R. Bercnji, "Fuzzy and Neural Control', Chapter 9 in An Introduction to Intelligent and Autonomous Control, Antsaklis and Passino Eds Kluner, May 1992. [3] M. Brown and C. Harris, "A Nonlinear Adaptive Controller: A Comparison between Fuzzy Logic and Neurocontrol', IMA Journal of Mathematical Control and Information (1991). [4} K. Uehara and M. Fujise, ' Learning of Fuzzy-Inference Criteria with Artificial Neural Network', Proceedings of the International Conference on Fuzzy Logic and Neural Networks {Iziika, Japan, July 20-24, 1990) pp. 193-198. [5] S. Nakanishi, T. Takagi, K. Unehara and Y. Gotoh, 'Self-organizing Fuzzy Controllers by Neural Networks', Proceedings of the International Conference on Fuzzy Logic and NeuralNetworks(Izuka, Japan, Jaly 20-24, 1990) pp. 187-193. [6} B. Kosko, Neural Networks and Fuzzy Systems. A Dynamical Approach to Machine Intelligence, Prentice Hall, Englewood Cliffs, NJ07632. [7] J. S. Judd, Neural Network Design and the Complexity of Learning, A Bradford Book ,The MIT Press, Cambrige, Massahusetts, London, England. [8] P. J. Antsaklis, Neural Networks in Control Systems, IEEE Systems Magazine, April 1990. [9] R. P. Lippman, 'An Introduction to Computing with Neural Nets', IEEE ASSP Magazin, April 1987. [10] R. Battiti, "First and Second Methods for Learning: Between Steepest Descent and Newton's Method', Neural Computation 4,141-166, Massahusetts Institute of Technology. 1992. [11] K. Watanabe and S. G. Tzafestas, Learning Algorithms for Neural Networks with the Kalman Filters', Journal of Intelligent and Robotic Systems 3: 305-319, 1990. [12] L. A. Zadch, Fuzzy Sets', Information and Control, pp. 338-353, 1965. [13] G. J. Klir, T. A. Folger, Fuzzy Sets, Uncertainty and Information, Prentice Hall, Englewood Cliffs, N. J. 1988.

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161

[14] Fllor Van Der Rhee, Hans R. Van Nauta Lemke, Jaap G. Dijkman, 'Knowledge Based Fuzzy Control of Systems', IEEE Trans, on Automatic Control, Vol. 35, NO. 2, February 1990. [15] J. M. Keller, R. R, Yager, H. Tahani, 'Neural Network Implementation of Fuzzy Logic', Fuzzy Sets and Systems 45 (1992), pp. 1-12, North-Holland. [16] C. C. Lee, "Fuzzy Logic in Control Systems: Fuzzy Logic Controllers, Part I, 11', IEEE Trans, on Systems, Man and Cybernetics, Vol. SMC-20, No. 2, March/April 1990. [17] M. Mizumoto, S. Fukami and K. Tanaka, 'Some methods of Fuzzy Reasoning', in: M. M. Gupta, R. K. Ragade and R. R. Yager, Eds., Advances in Fuzzy Set Theory and /4/ip7/ca/«7i2s (North Holland, Amsterdam, 1979) pp. 117-126, [18] S. G. Tzafestas, 'Fuzzy Expert Control: Recent Results with Applications to Robotic Systems', Advanced Study Institute oa knowledge Based Robot Cootrol, France, 1988. [19] S- Tzafestas and N. P. Papanikolopoulos, 'Incremental Fuzzy Expert PID Control", IEEE Trans, on Industrial Electronics, Vol. 37, NO. 5, October 1990. [20] T. J. Procyk and E. H. Mamdani, 'A linguistic self-organizing process controller,' Automatica, vol. 15, pp. 15-30, 1979. [21] M. Braae and D. A. Rutherford, 'Theoretical and linguistic aspects of the fuzzy logic controller,' Automatica, vol. 15, pp. 553-577, 1979. [22] P. J. Macvicar-Whelan, "Fuzzy sets for man-machine interaction,' Int. J. Man-Mach. Studies, vol. 8, pp. 687-697, 1976.

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PART 3

FUZZY CONTROLLERS

CHAPTER 8

DEMYSTIFICATION OF FUZZY CONTROL R. Jager, H.B. Verbruggen and P.M. Bruijn Control Laboratory Department of Electrical Engineering Delft University of Technology P.O. Box 5031, 2600 GA Delft, The Netherlands

A b s t r a c t . This contribution addresses a "down-to-earth" approach towards fuzzy control. A description is done of the basic operations in fuzzy controllers. Besides the global description critical remarks are placed at those pMts which are necessary for a correct understanding of the working of fu2zy controllers. The application of fuzzy logic in control can be regarded as a concept for the design of non-linear controllers, which is even understandable for non-control engineers.

1

Introduction

Fuzzy control has become a popular topic in t h e control engineering community. T h e proliferation of fuzzy control m e t h o d s get really underway by t h e introduction of fuzzy control in consumer products by t h e Japanese electronic industry. Also, applications are reported in the process industry (Aoki et al, 1990; Yu et al, 1990) of which the best known are in the cement industry (Haspel, 1991; Ostergaard, 1990; Flintham, 1991) Many instrumentation firms offer fuzzy hard- and software as implementation tools for fuzzy control, like for example, Siemens, Yokogawa, Inform, Togai, etc. W h a t is the reason t h a t fuzzy control methods became so popular in control engineering? Most modern production processes require more advanced control. Frequent changes in production rate, product mix and production procedures and recipes, give rise to a completely different situation in which linearizations around operating conditions are hardly possible, because many process models are incomplete, time-varying and highly non-linear. 165

S.G. Tzafestas and A.N. Venelsanopoulos (eds.) Fuzzy Reasoning in Information, Decision and Control Systems, 165-197. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.

166

Most control algorithms, however, axe based on Hnear control theory and too much relying on this approach leads to a dead end for their use in modern production methods. Moreover, it is clear that the present state of t h e analysis and synthesis of non-lineax controllers based on non-Iineax system theory is still i m m a t u r e and does not provide the solutions needed in industry. So, t h e control engineering community was waiting for new methods which could alleviate the problems which have to be solved in process industry. Fuzzy control algorithms could be very helpful, because of t h e following two reasons. • They allow the use of linguistic expressions to describe the behaviour of the system or the control actions. Using this property it is possible to i m i t a t e the actions of operators. • They are inherently non-linear and are thus able to perform control actions, which are not possible with a purely linear approach.

These properties alow fuzzy control to be brought into action in many control problems. However, some people think t h a t fuzzy control is t h e panacea for everything t h a t is difficult to control. It is true t h a t some authors and advocates of fuzzy control have stimulated this feeling by only indicating the advantages of fuzzy control and mystifying the real problems arising when applying fuzzy control. There is a lack of knowledge of the relationship between fuzzy control algorithms Eind non-linear controllers. Furthermore, it is not clear how the structure and parameters of a fuzzy controller should be chosen in a given situation. This subject is seldomly described by the users and advocates of fuzzy control and t h a t is the reason t h a t people not involved in this field have the feeling t h a t t h e method is mystical, unreliable, lacks mathematical rigorness and is a box full of tricks. In this contribution we want to show the real face of fuzzy control, showing the strength of this m e t h o d in comparison to other methods, but also the weak points and unsolved problems. We will confine this contribution to fuzzy control in which t h e controller is part of a closed loop system, as shown in figure l a . T h e use of fuzzy algorithms for supervision (see figure l b ) and decision meiking will not be treated in this contribution. A good overview of fuzzy controllers has been given by Lee (1990a,b). The first three sections of our contribution are concerned with the basic parts of a fuzzy controller as shown in figure 2. First the fuzzification of the inputs of the fuzzy

167

FS r

FC

u

1

1 y

process

y

+ (a) Fuzzy controller (FC).

(b) Fuzzy supervisor (FS).

F i g u r e 1: Fuzzy control

loops.

controller will be described, dealing with the choice, number and (relative) position, of membership functions. T h e second section will discuss the inference part of fuzzy controllers. T h e a i m of section 4 is t h e comparison a n d evailuation of defuzzification methods. Next the relation between fuzzy control and lineaj control is described. T h e last section will be used for conclusions and discussion.

rule base

n

X

rule of inference

u^

'S -a

F i g u r e 2: Schematic

2

representation

of fuzzy

controller.

Fuzzification

Although in m a n y applications of fuzzy control the influence of the choice of m e m bership fimctions is hardly discussed, it is an important issue. Mainly because the

168 n u m b e r and shapes of those membership functions a n d how they cire related to Ccich other, as well as the number of overlapping membership functions, determine the resulting controller o u t p u t as function of the controller inputs of the fuzzy controller. In this section a centre-of-gravity oriented defuzzification (see section 4) method is assumed. T h e first three subsections will deal with t h e fuzzification of the inputs. T h e last subsection will address "fuzzification" of the universe of discourse of the o u t p u t .

2.1

Number and positions of membership functions

W h e n designing a fuzzy controller one of the first questions which arise, after having chosen the inputs, is how many membership functions are needed eind how the membership functions should be divided on the universe of discourse of the inputs.

(a) Linear division of membership functions on universe of discourse.

(b) Non-linear division of membership functions on universe of discourse.

F i g u r e 3: Commonly used basic set of m,embership functions verse of discourse (meaning of abbreviations: (N)egative, (A)bout, B(ig), (M)edium, (S)mall and (Z)ero).

on a uni(P)osttive,

In figure 3 more or less standard types of universes of discourse of controller inputs

169 cire shown. Whether to use a linear (3a), a non-linear (3b) or an other set of membership functions on the universe of discourse of an input depends on the problem to deal with. T h e higher the density of membership functions on a certain part of the universe of discourse, the more complex the controller o u t p u t as function of the controller inputs can be obtained. Actually, the choice of the number of membership functions and how those membership functions are divided over t h e universes of discourse requires knowledge about how t h e controller o u t p u t should b e related t o the controller inputs. For example, designing a fuzzy controller for controlling a non-linear process requires knowledge about the non-linearity of the process. There are no rules of t h u m b to choose the number and positions of the membership functions, and few people realize t h a t this is a problem. T h e number and positions of the membership functions define the control hypersurface and proper design of this control hypersurface requires process knowledge. When this process knowledge exist in the form of operator expertise then one has to deal with the knowledge acquisition problem known from the field of expert systems.

2,2

Overlapping membership functions

It can be shown that having more t h a n two overlapping membership functions will lead to a sort of "smoothening" of the control hypersurface, t h a t would result from having only two overlapping membership functions. In figure 4a the smoothening effect on the o u t p u t u as function of the input x is shown, which can occur when there exist more t h a n two overlapping fuzzy sets on a universe of discourse. Open marks represent the numerical results of individual fuzzy rules in case of no more t h a n two overlapping membership functions. T h e solid marks represent the numerical output of the fuzzy controller when the support value of one rule equals 1 and there exist more t h a n two overlapping membership functions. T h e gray coloured curve in figure 4a represents t h e controller output u as function of the controller input x in case no more t h a n two overlapping membership functions are used. T h e black curve represent t h e controller o u t p u t due to more t h a n two overlapping membership functions. As figure 4a shows, the results of individual fuzzy rules are not part of the black curve: fuzzy rules interfere with the results of individual fuzzy rules. This interference due t o more t h a n two overlapping membership functions leads to effects on the controller hypersurface, which are not e£isy to see a priori. Using, for example, Gaussian membership function will introduce this effect. This interference

170

can best be shown by having a set of rules in which t h e conclusion of one m l e is changed. In figure 4b one can see the effect of changing t h e concbision of one rule (marked by an axrow): it also influences the numerical result of other rules. O n the other hand, when Gaussian membership functions axe used and t h e fuzzy sets are composed in such a way t h a t the overlapping of non-adjacent membership fiinctions is neglectable, this interference between rules and thus t h e "smoothening" of the control liypersurfacc will also be neglectable.

(a) Smoothing of controller output u{x) as function of controller input u.

F i g u r e 4: Possible bership functions.

2.3

(b) Influence of changing conclusion of one fuzzy mle controller output.

effects in case of more than two overlapping

mem-

Shape of membersMp functions

As discussed in the previous subsection, more t h a n two overlapping membership functions will result in some non-trivial characteristics of the fuzzy controller. T h a t Iccives us to the standard continuous and discontimions (sometimes referred t o as "straight") membership functions. T h e straight membership functions, like t h e trapezoidal shaped membership function (subsection 4.2, figure 8c,d), represent t h e linear approach, t h e continuous membership functions, lilse the bell-shaped m e m bership function (subsection 4.2, figure 8a,b), represent the non4inear approach.

171

W h e n the output of t h e fuzzy controller is regarded as a function of the inputs, which is determined by the interpolation between the numerical results of the individual fuzzy rules, t h a n this interpolation is non-linear in case of t h e continuous membership functions. By using non-linear membership functions a non-linear characteristic is introduced in the fuzzy controller, which is not a trivial CEise. Clearly this non-linear interpolation can provide the control surface wanted, but this is, in our opinion, not according to the basic idea of a fuzzy controller: the non-linearity of the fuzzy controller should be defined by the fuzzy rules, and thus by t h e number and positions of the membership functions, because these rules are the representation of our process knowledge. Also the "widths" of the membership functions are relevant: the width of the pEirt where the membership function equals 1 and t h e widths of the parts where the membership function is between 0 and 1. T h e effects on the controller output due to the shapes of the last mentioned peirts, however, are hard to determine.

2.4

Membership functions for the output

It is possible t h a t the output of a fuzzy controller also ha.s a universe of discourse on which membership functions are defined. This, however, is not always the case as will be show in section 3.2. In case membership functions are defined for the controller o u t p u t , this is normally done using a set of equidistant chosen membership functions on the universe of discourse. It is hard to make any general remark concerning the "fuzzification" of the output, because its effect depends for a great deal on the fuzzy rules. Because of this and the interpolative character of a fuzzy controller, a chosen set of membership functions do not limit the resolution of t h e controller o u t p u t , assuming a centre-of-gravity oriented defuzzification method (see also section 4). T h e number and positions of the membership functions on the universe of discourse of the output, in combination with t h e fuzzy rules and the universes of discourse of t h e controller inputs, should be chosen according to the control hypersurface desired.

3

Rules, operators and inference

In this section we will address fuzzy rules, the connectives used in fuzzy rules axid fuzzy inference of t h e rules. First, we will start with t h e logical operators to implement the connectives. Secondly the fuzzy rules are described and after t h a t the inference of the fuzzy rules is discussed.

172

3.1

Connectives

To express knowledge by means of fuzzy rules one needs logical connectives. T h e metin logical connectives used in standard fuzzy controllers are: and cind then. Of course connectives like or and not are possible, but not explicitly necessary. Sentence connectives are mentioned in literature like else, for example by Harris and Moore (1989), and also, for excimple by Lee (1990b). We will not go into detail about the meaning of these sentence connectives, which are used to combine the fuzzy rules. For implementation of the and connective a T-norm (Schweizer & Sklar, 1963) is used. T h e conditions which a T-norm, for applying intersection or conjunction, must satisfy are: T-1

T(a,l) = a

T-2

T{a, b) < T ( c , d), whenever a < c, b < d

T-3

T{a,b)

=

T-4

T{T(a,

b), c) = T(a, T(b, c))

T{b,a)

In this paper ^ will be used to denote a T-norm^. Some examples of T-norms axe: m m ( a , 6 ) (Zadeh), max(a + b — 1,0) (Lukasiewicz), ab (probability theory), etc. Several authors have defined families of T-norms (and -conorms), which have parameters to vary the function of the T-norm (or -conorm). For example, t h e T-norm family according to Dubois and Prade (Klir & Folger, 1988):

fd&p,e(a,<> =

ab , , .. max(a,b,(fj

(la

with 6(z [0,1]. When 6 equals 0, <
S-I

S(a,0) = a

A S-norm will be denoted in this paper by \l. Examples of S-norms are: max(a,b), min[a-\-b, 1), o-|-6 — ab, etc. As with the T-norm there exist a lot more possibilities.

173 often combined in a family of T-norms and -conorms. For example, the S-norm related to ( l a ) is defined as: a + 6 — a6 — m w ( a , 6 , 1 — ^) max{\

,

— a,l — 0,6)

This S-norm td&p,e acts as max(a, b) when 6 equals 0 and acts as a + b — ab from probability theory, when 6 equals 1. For more detailed information on T-norms emd -conorms we refer to Klir and Folger (1988), Buckley and Siller (1987) and Turksen (1986). Although not further used in this paper, but for completeness, t h e minimal conditions for the complement, used for implementation of the connective not, are (Hellendoorn, 1990): C-1

C(0) = 1

C-2

C{a) > C(b), whenever a
C-3

C(C{a))

= a

For t h e complement function normally 1 — a is used. The next section will use the T-norm and -conorm in the compositional rule of inference.

3.2

Fuzzy rules

In fuzzy control rules are used to represent the knowledge implemented in the controller. In this section we will not discuss the type of knowledge the fuzzy rules represent, like state evaluation or object evaluation fuzzy rules (Lee, 1990b), because this reflects more the application of fuzzy rules than t h e form. Here we will address only the form of fuzzy rules and one can mainly distinguish two types: rules with "symbolic" consequents and rules with "numerical" consequents, from now on referred to as Mamdani- and Sugeno-rules, respectively, eind discussed in t h e following subsections.

3.2.1

M a m d a n i - t y p e fuzzy rules

This type of rules was used in the first applications of fuzzy control (Assilian, 1974; Mamdani, 1974; Mamdani & Assilian, 1975) and has the following general form:

,

174 TJ: if Xi is X( and • • • Eind XN^ is X^j

then u is [/•'

with: N3. Xi Xi u U^

: : : : :

n u m b e r of controller inputs i'*' input of controller fuzzy label for i"* input in j " * rule o u t p u t of controller fuzzy label for controller output in j ' ' ' rule

T h e consequent of this type of rules is a symbolic one, for example, "controller o u t p u t is big". Theoretically a fuzzy rule can be represented by a relation Rj, which is the cartesian product of the conditions in t h e premise of a fuzzy rule and the consequent of the fuzzy rule:

R^ = {Xix...xXi,JxU^

(2a)

We can write (2a) as a multi-dimensionaJ fuzzy set:

^iR, (KI , • • •, xjv,, u) = { A l^x' (==0} A t^u- (w)

(2b)

Figure 5 shows an examiple of a 2-diniensional fuzzy set representing a fuzzy rule, using the min-operator for conjunction. Section 3.3 will explain how these fuzzy rules are combined and used for fuzzy inference. Using the Mamdani-types of rules in a fuzzy controller will result in a fuzzy controller o u t p u t as result of the fuzzy inference, which has to be defuzzified to obtain a numerical controller output. Defuzzification methods are addressed in section 4.

3.2.2

S u g e n o - t y p e o f f u z z y rules

T h e general form of the other rule type, referred t o as Sugeno-rules, because of the introduction of those rules by TaJcagi and Sugeno (1983) and further exploited by Sugeno and co-workers, is as follows;

rji if Xi is X( and • • • and x^^ is X^^ then u is /•'{xi, • • • , x^^)

175

F i g u r e 5: Representation means of fuzzy relation.

of a fuzzy rule (if x is

.then

•) by

which shows t h a t t h e consequent of a fuzzy rule is a function (/J) of t h e controller inputs. A linear function in the consequent will simplify this rule type to:

Tf. if Si is X( and • • - and xjv, is X}f^ t h e n « is ofxi -f . . . + a^^asiv. + V>

with constant parameters af and ¥ and which can b e even more simplified when t h e parameters af are 0, resulting in a constant numerical consequent 6^'. Successful use of this t y p e of fuzzy rule in t h e control of a model car was reported by Siigeno and Nishida (1985) and Sugeno and Murakami (1985). T h e "fuzzy" o u t p u t of a fuzzy controller using this t y p e of fuzzy rules is in fact a set of weighted numerical o u t p u t s . A simple defuzzification results in the final controller o u t p u t . A set of Sugeno^rules can be seen as a set of (non-)hnear controllers, between which is interpolated by means of fuzzy inference. This possible connection with conventional control techniques will be addressed in section 5. Although one can distinguish the Mamdani-rules from the Sugeno-rules, in practical set-ups the Mamdani-type of rules are quite often similar to Sngeno-mles with constant consequent, for example, see Batur and Kasparian (1991), Harris and Moore (1989) and Matsuoka (1991). This is due to t h e defuzzifi cation m e t h o d used to defuzzify the fuzzy output of t h e controller: fuzzy-mean or centroid. See section 4 for details on defuzzification methods.

176

3.3

Fuzzy inference

T h e application fuzzy relation R fuzzy set, which fuzzy rules (see

of fuzzy inference is based on t h e idea of relational implications. A representing a fuzzy controller is represented by a multi-dimensional is a combination (union) of the fuzzy sets representing the individual section 3.2):

R=\:)Ri

(3a)

in which a "starred" union operator is used to express, as in subsection 3.1, a general form of the operator. W h e n using fuzzy sets: Nr

fJ-R{xi,••• ,XN,,u)

= ^ fiRjixi,---

,XN,,u)

(3b)

3=1

where Nr is the number of rules. compositional rule of inference:

A fuzzy output is obtained by applying t h e

U^X,o{X2o{---iXr,,oR)...))

(4a)

where Xi axe fuzzifications of the controller inputs Xi and U is the fuzzy o u t p u t . More commonly the compositional rule of inference is written as: U = Xo

R

(4b)

where X is the cartesiaji product of the fuzzy inputs Xi and using fuzzy sets: t^uiu) =

V

{it^x{3:i,---,XN,)/\fJ.Rixi,---,XN,,u))}

(5a)

IJ'xixu---

,XNJ

=MJ:i(a;i)/\---A/iXN,(«wJ

(5b)

with:

In practical applications however, usually a more efficient m e t h o d is used (Harris & Moore, 1989). This method is based on the assumption t h a t the fuzzification of an

177 input can be represented by a singleton, which is often true in control, because of t h e fact t h a t t h e controller inputs are normally "crisp" measurements. Because t h e fuzzification of the inputs can be represented by singletons, the same fuzzy o u t p u t can be obtained by determining for every fuzzy rule Rj the individual fuzzy output: AT.

where one should note t h a t here Xi(t) is used instead of Xi as in (5). This means t h a t the left T-norm in (6) operates on membership values and not on membership functions as t h e right T-norm does. In figure 6 this practical way of fuzzy reasoning is shown schematically, where the rnin-operator is used as T-norm. Combination of those individual fuzzy outputs by applying a union results in the complete fuzzy o u t p u t of the system:

IJ.u{u) = V nui{u)

(7)

3=1

T h e result of the union of the results of figure 6 is shown in figure 7. A max-operator is used for t h e union. T h e most commonly used inference rule is the maa;-mm-rule (the one used in the examples in figure 6 and 7). Also frequently used is the max-prod type of inference. Fuzzy inference using non-lineax operators (others thaji the product and summation operator for T-norm and -conorm, respectivily) results in a non-linear control hypersurface. Even when the fuzzy rules form a consistent, complete and "linear" rule base, the control hypersurface will b e non-linear. This would not be the case using t h e sum-prod-rule of inference, consisting of "linear" operators. T h e non-lineaxity of the control hypersurface is completely determined by the logical operators and not by the fuzzy rules, which, in our opinion, should define t h e non-linearity of the control hypersurface. Section 5.2 will show this in more detail in case of a fuzzy PID-controller. Several defuzzification methods are available to translate the fuzzy output of the controller into a crisp, numerical output value. Defuzzification will be discussed in t h e next section.

178

(a) If xi is Ai and 12 is B2 then u is Ci.

If Xi is A2 and X2 is Bi then u is €'2-

F i g u r e 6: Schetnatical representation min-operaior for and^connective and

of practical fuzzy reasoning implication.

using

179

1.0 J''

0.5

^1

y

G2

Figure 7: Combination of fuzzy outputs of individual fuzzy rules, as shown tn figure 6, using maa-opemtor.

4

Defuzziflcation

In this section the role of defuzzification methods in fuzzy controllers is addressed. A closer look at the influence of this specific part of the fuzzy controller is worthwhile, because from hterature several defuzzification methods are known. The best known defuzziication methods are (Harris fe Moore, 1989; Jager et al, 1992; Larkiii, 1985; Pedrycz, 1989; Hellendoorn, 1993); centre^otarea (coa) or centre-of-gravity (cog)', fazzy-mean (fm) or centroid (ctrd), weighted-fuzzy-mean (wfm), indexed versions of ^those (id/z) and mean-otmaxima (mom) defuzzification method. Adaptive or adjustable defuzzification methods become more popular (Yager & Filev, 1993), but will aot be addressed in this contribution. The defuzzifier of a fuzzy controller can deteriorate a great deal the results achieved by the inference of fuzzy rules. It appears that the application of a specific defuzzification method can reduce completely the fuzzy character of the controller, as will be shown in subsection 4.2.

4.1

Defuzzification methods

This section will describe the basic defuzzification methods and related methods. Probably the best known method is the centre-of-area defuzzification method (coa),

180 which is defined by; coa{U)

/

TOO

Hu(u)du^

I

iJ,u{u)du

(8a)

resulting in a numerical value u = u^oa which divides the integral of the fuzzy output in two equal parts. This way of calculation is normally not applicable in practical situations. T h e centre line u — Ucoa, which divides the area under the fuzzy output function caji be approximated by calculating the centre of "gravity", which explains the alternative name, centre-of-gravity (cog) method: JV,

cog(C/) = i ^ i

(8b)

where iV, is the number of quantisations used for the discretisation of the fuzzy output ^i/(u). T h e fuzzy-mean method (fm) uses only, pre-calculated, crisp representations of the individual fuzzy sets on the universe of discourse of the controller output and is in fact the centre-of-gravity using as less quantisations as possible:

fm(C/) = '^^

(8c)

k=\

where N^ is the number of fuzzy sets defined on the universe of discourse of the controller o u t p u t , 7^ denote the support values for membership function Uu as result of the inference within the controller and u^ represents the defuzzification of U^. The support values 7fc are determined by the conjunction of t h e condition part of the fuzzy rules and therefore the fuzzy-mean defuzzification method is very suitable in case the calculation load of the fuzzy controller should be small (see subsection 3.3). In figure 6 the support values are t h e values by which the fuzzy outputs Ci and C2 are limited (in case of m i n i m u m as T-norm for the implication). These values are calculated in figure 6 by taking t h e m i n i m u m of the membership values in the premise of the fuzzy rules. Because only crisp representations are used of

181 t h e fuzzy sets defined on the universe of discourse of the controller o u t p u t , this defuzzification method treats Mcimdani-type of rules as Sugeno-type of rules with a constant consequent. A weighted version of t h e fuzzy-mean m e t h o d (wfm) is defined by:

wfm(C/) == ^^L

(8d)

in which Wj are weights assigned to the membership functions on the universe of discourse of the controller output to emphasize certain membership functions (Jager et al., 1992). Indexed defuzzification methods {idfz^ are defined by: Ufz{U,ai)

= dfz{a-cni{U,ai))

(8e)

and are used to discriminate part of a fuzzy o u t p u t of which t h e membership values are below a certain threshold a,: the defuzzification m e t h o d is only applied on the part(s) of the fuzzy output, which has a membership vaJue greater (or equal) 0:^. This discrimination provides a way to let t h e defuzzification of the fuzzy o u t p u t tend towards the m a x i m a of the fuzzy output. An extreme form of this will result in t h e mean-of-maxima defuzzification method (m.om), because t h e mean-of-maxima m e t h o d uses only the m a x i m a of t h e fuzzy o u t p u t : mom(C/) = icog(C/, hgt(C/))

where hgi{U)

4.2

(8f)

represents the height ( m a x i m u m value) of the fuzzy output.

Comparison and evaluation

In this subsection we will take a closer look at the several defuzzification methods as described in the previous subsection. Comparisons will be made, advantages and disadvantages will be discussed. First t h e mean-of-maxima method is evaluated; after t h a t a comparison is made with t h e centre-of-area related defuzzification methods.

182 Although t h e mean-of-maxima m e t h o d is an obvious m e t h o d for defuzzification (Dubois &: Prade, 1980), applying it in fuzzy control reduces t h e "fuzziness" of the controller completely in m a n y cases. In case the mean-of-maxima m e t h o d is used for defuzzification of the fuzzy output, the membership functions describing the inputs can even be chosen as crisp (classical) sets, without having any effect on the numerical o u t p u t . Suppose the fuzzy controller satisfied the assumptions given in t h e introduction. When the chosen T-norm satisfies, besides the conditions T - 1 , T-2, T-3 and T-4 as stated in section 3.1, also a stronger condition for T-2, namely:

T-2

T{a,b)

< T{c,d),

whenever a
b
£ind it is assumed t h a t the membership values do not have values equal to ~, then there exists only one dominant fuzzy rule in the rule base. This domincint rule will completely determine the numerical output of the fuzzy controller in case the chosen S-norm for combining t h e individual fuzzy conclusions of the fuzzy rules (union) satisfies besides the conditions S-1, S-3 and S-4 a stronger condition for S-2, namely:

S'-2

S{a,b)

< S{c,d),

whenever a < c, b < d

So every combination of T- and S-norms satisfying the above listed conditions, for excimple the max- and min-operators and t h e product operator, will result in the same numerical output of the fuzzy controller. This means, for example, t h a t redefinition of the shape of the fuzzy sets defined for the inputs in a way as is done in figure 8 has no effect at all on t h e numerical o u t p u t . In case we consider also the cases in which more t h a n one fuzzy rule are equally dominant (two or more membership values are equal to i ) , t h a n the (finite) number of possible numerical values for the o u t p u t is still restricted by:

zJ

r, :T?^ _ M „fj ;n\{N^ - ^n)\ n I ""^

(^^

Another disadvantage of the mean-of-maxima m e t h o d is the fact t h a t non-symmetrical fuzzy sets defined for the o u t p u t can result in undesired behaviour. In figure 9 two situations are depicted which point out t h a t a shift of t h e fuzzy o u t p u t possibly results in a shift in the opposite direction of the numerical o u t p u t .

183 1' L . . .^. . . .

/ 0

1'

\

y V (a) A-function.

0

r 1

/ \

*•

0

(b) U-function.

F i g u r e 8: Different

>

0/

\

m

Triangular,

shapes of membership

\:

Trapezoidal. functions.

/*£/(")

(«)

/H/(ti)

(b)

F i g u r e §: Undcsired result of mean-of-mamima non-symmetrical fuzzy sets: shift of numerical shift in fuzzy output.

method •in case of output opposite to

184

it

0

200

400

600

800

lOOO

1200

1400

1600

1800

2000

1200

1400

1600

1800

2000

1400

1600

1800

2000

1400

1600

1800

20O0

tune(s)

0

200

400

600

800

1000 time (s)

(a) Mean-of-maxima defuzzification.

a I ^

Oh

0

200

400

600

800

0

200

400

600

800

1000 1200 time (s)

lOOO

1200

time (s)

(b) Centre-of-gravity defuzzification.

F i g u r e 10: Experimental results with fuzzy control using different defuzzification methods.

185 T a b l e 1: Number and types of operations in centre-of-gratrity related defuzzification methods, with Nq > 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 <— Y

\sB'

Where the variables X,Y take values in universes of discourse U,V respectively and A,A',B,B'

are (possibly inexact) description of the linguistic variables X

a n d Y, which are approximated by fuzzy sets over U,U,V,V respectively.

In extended fuzzy reasoning as suggested by Mizumoto [23], we make inferences of t h e form

202 P:if Xi is Ai and X2 is A2 and ... X„ is A„ then Y is B q:Xi is i4j and Xj is ^'2 and ...X„ is A'^

(concl.) r <— y is B'

Where XJ5 are variables taking values in universes of discourse U,; i = l ,2, .... n and Y is a variable taking values in Y.A; and A'^ are fuzzy subsets of {/, : i = 1, 2, ...., n. B and B' are fuzzy subsets of V. The above fuzzy implication inference rules are known as generalized modus ponens (GMP). The propositions above the line are the premises; and the proposition below the line is the conclusion. The criteria of inferences proposed by Zadeh [22] are summarized in Table 1. We note that if a causal relation between " x is A" and "y is B" is not strong in a fuzzy implication, the satisfaction of criterion 2-2 and criterion 3-2 is allowed. Criterion 4-2 is interpreted as: if x is A then y is B, else y is not B. Although this relation is not valid in formal logic, we often make such an interpretation in everyday reasoning.

2.1

Fuzzy Implication Functions:

Following Zadeh's [20] introduction of the compositional rule of inference in approximate reasoning, a number of researchers have proposed various implication functions in which the antecedents and consequents contain fuzzy variables. After a short review of triangular norms and triangular co-norms, [60] we shall state some fuzzy implication functions, which are often employed in a FLC and are commonly found in the literature.

203 Definition 1:

The triangular norm * is a two-place function from [0, l] x |0, Ij

to [0,1] which includes intersection, algebraic bounded product and drastic product. The greatest triangular norm is the intersection and the least one is the drastic product. The operations associated with triangular norms are defined for all x,y € [0,1]:

X t\y — min(a;,y)

intersection algebraic product

x.y = xy

bounded product

X 0 y = max(0, x + y — 1) X y= 1

drastic product

xny

y X= 1 0 x,y < 1.

Definition 2: The triangular co-norms + is a two- place function from (0, l] x (0, l](o[0,1] which includes union, algebraic sum, bounded sum drastic sum and disjoint sum. The operations associated with triangular co-norms are defined for all X, y 6 (0,1]

union

x V y = max(x, y)

algebraic sum

x^y

= x + y — xy

bounded sum

x ® y = min(l,x + y) X y= 0

drastic sum

xl±iy=:< y x = Q 1

disjoint sum

x,y>0

xAy = max{77M'n(i, 1 — y),min(l — x,y)}

The triangular norms are employed for defining conjunctions in approximate reasoning, while the triangular co-norms serve the same role for disjunctions. A

204 fuzzy control rule, "if x is A then v is B" is represented by a fuzzy implication function and is denoted by A —*• B , where A and B are fuzzy sets in universes U a n d V with membership functions / i ^ and fis respectively.

Based on the above definitions many fuzzy implicatioii functions may be generated by employing the triangular norms and co-norms. For example M a m dani's min-fuzzy implication, Re is obtained if the intersection o p e r a t o r is used, Larsen's p r o d u c t fuzzy implication Rp is obtained if the algebraic p r o d u c t is used.

F u r t h e r m o r e , fi^^ and R^^ are obtained if the b o u n d e d p r o d u c t and

the drastic product are used respectively. T h e following fuzzy implications are often a d o p t e d in the design of a FLC [l] [23] [60] [81].

(i) Arithmetic rule of fuzzy implication |Zadeh]: MR, = [notA X v) ® (U X B) = Jy^^ 1 A (1 - MA(W) +

fJ-B{u))/(u,v)

(ii) M a x i m u m rule of fuzzy implication [Zadehj: Mfl„ = [notAx

B ) U ( ^ X V) = / U X V I M A I " ) AMfl(t;)) V (1 - ^ ^ ( u ) ) / ( u , v)

(iii) Min-operation rule of fuzzy implication [Mamdani]: tj.j{^. - A X B ^ J(j^v t^A(u) A

^IB(V)/{U,V).

(iv) P r o d u c t operation rule of fuzzy implication [Larsen]: HR^ = A X B ^

Ji,^yfj.A{u)fJ.B{v)/{u,v).

(v) S t a n d a r d sequence fuzzy implication: tiR, =AxV-^UxB^ where

Juxvit^A(u)

>

tiB{v))/iu,v)

205

0

HA(U)

>

HB{V)

(vi) Goguen's fuzzy implication: fiR^=AxV-^UxB

= Ju^y{fj.A{u) »

fiB{v))/(u,v)

Where

fJ,A(u} »

fJ.B(V) = <

(vii) Boolean fuzzy implication: tj.R^ = (notAx

V)U{U

X B) =/y^v,(l - ^ A ( « ) ) V {MB(v))/(u,t;)

By employing the intuitive criteria of Table 1 we list in Table II how well a fuzzy implication function satisfies them.

Although nn^, and fiRp do not have a well-defined logical structure the results tabulated in Table II indicate that they are well suited for approximate reeisoning especially for the generalized modus ponens. fiR^ has a logical structure which is similar to iiR^-tiR^ is based on the implication rule in Lukasiewicz's logic. However, HR^ and ILR^ are not well suited for approximate reasoning since the inferred consequences do not always fit our intuition. Furthermore, for multiple-valued logical systems JIR^ and /x^^ have significant shortcomings. Over-all ^IR^ yields reasonable results and thus constitutes an appropriate choice for use in approximate reasoning.

206

2.2

Method of Redefinition [24]

If the membership functions of fuzzy sets A and B satisfy the following conditions:

{(IA{U) \ueU}^

{fJ.B[u)

\ueV},

3« e u,fi4u) = o,3u' e u,^4u') = i, ^v e

V,I^B{V)

= 0,3t;' e V,tiB{v'} = 1,

then the combination of the max-min compositional rule of inference and the fuzzy relation /x/j, satisfies the following criteria which are the subset of the criteria given in Table I. For the rule if X is A then Y is B:

(i) Known X is A, Y should be concluded as B. (ii) Known X is very A, Y should be concluded as very B. (iii) Known X is more or less A, Y should be concluded as more or less B. (iv) Known X is not A, Y should be concluded as unknown.

3.

Steps involved in t h e design of FLC

The aim of this section is to introduce the basic ideas underlying the design of FLC. In Figure-1 we have shown a the basic configuration of a closed-loop system associated with a fuzzy logic controller.

207 Non(uizy error Input to FLC

4-

<^

JL

Nonfuzzy output of F L C

A

FLC

r i g . 1 : Closed-loop

' u z z y logic

System

controller

In Figure-2 we have shown the fundamental components which construct the FLC.

Now, in the following we will describe the methods through which we can realise the individual components of FLC and the functional involvements of the components of FLC.

3.1

Fuzzification of the error input

In most of the fuzzy control applications, the error inputs to FLC are nonfuzzy in nature. But the data manipulation in a FLC is based on fuzzy set theory. Hence, fuzzification of the error input to FLC is necessary. In the following we

208 r " ^^. 1

NonKJizy input

;

J Knowledge Base

, Fu22)fiC0»'0n the

prror

of

tnpu!

-^

^^

L

Fuzzy Inftrence using fhe method of opproximote reosomng

Nonhjzty o j t p j f of F

1 Defuzzifico* on of Ihe t u n y inference

1 f 1

.

Fu2 2y ioqic controMtr ( F L O

F i g ? : Esttnliol

compontnis of

FLC.

briefly describe some existing tools for fuzzification.

(i) An error input, Co, is assumed to be a fuzzy set A with the membership function

/XA(*O)

equal to zero except at the point e„ at which ^^(^0) equals

to one [60] (ii) We may assign each error input to a fit vector of length L where the ith fit or fuzzy unit [25] equals the value of the ith fuzzy set at the input value. For instance, let us consider the Fig-3 which provides the library of overlapping fuzzy set values defined on a universe of discourse. let the error inputs be —6,2,3.9. Then from the library of fuzzy set we obtain the fit vectors of length 5 as follows:

209

u. 1

/ 0

/ 10

Figa.Librar/

-6 ^ 2

of o v e r l o p p i n ^

fuzzy

sets-

(0,1,0,0,0)

=> (0,0,0, .7, .7)

3.9 => (0,0,0,.1,1) (iii) First the observed data are disturbed by random noise. Then the probabilistic data are converted into fuzzy numbers, i.e. fuzzy (possibilistic) data. In [26] [27] an isosceles triangle is chosen to be the fuzzification function. The vertex of this triangle corresponds to the mean value of a data set while the base is twice the standard deviation of the data set. In [28] Dubois and Prade define a bijective transformation which transforms a probability measure into a possibility measure using the concept of the degree of necessity. (iv) Sometimes observed data are precise as well as imprecise in some statistical sense. Such cases can be handled by hybrid number [27] which involve

210

both uncertainty (fuzzy numbers) and randomness (random numbers).

3.2

Knovs^ledge B a s e

The knowledge base of a FLC has two components; namely a data base and a fuzzy rule base. We shall consider some issues relating to the data base and rule base in the following subsections.

3.2.1

D a t a base

In the following we discuss some of the important aspects relating to the construction of the data base in a FLC.

(i) Discretization of a universe of discourse: Discretization is often referred to as quantization which discretizes a universe into a certain number of segments (Quantization levels). Each segment is labeled as a generic element, and forms a discrete universe. A fuzzy set is then defined by assigning grade of membership values to each generic element of the new discrete universe. An examples of discretization is shown in Table -III. The choice of quantization levels has an essential influence on how fine a control can be obtained. For example, if a universe is quantized for every five units of measurement instead of ten units, then the controller is twice as sensitive to the observed variables (60)

211

(ii) Normalization of a universe of discourse : The normalization of a universe requires a discretization of the universe of discourse into a finite number of segments. Each segment is mapped into a suitable segment of the normalized universe. Table - IV represents an example of normalization. In this setting a fuzzy set is defined by assigning an explicit function to its membership function. (iii) Membership function of a primary fuzzy set: Basically there are tw^o approaches to define fuzzy set. In case the universe of discourse is discrete we follow numerical definition whereas we follow functional definition if the universe of discourse is continuous. (a) Numerical

definition:

In this case the grade of membership function of a fuzzy set is represented as a vector of members whose dimension depends on the degree of discretization. As shown in Table - III, the membership function of each primary fuzzy set has the form of

where a = [0.4,0.6,1.0,0.8,0.4]. (b) Functional

definition—

In this case the membership function of a fuzzy set is represented by a functional form, namely bell-shaped function, triangle shaped function, trapezoid shaped function etc. Table - IV re presents an example of a functional definition expressed as,

;.K)=exp{^^^i=^}

212

For any practical purpose we may either use numerical definitions or functional definition to assign the grades of membership to the primary fuzzy sets. The choice of grades of membership is based on the subjective criteria of the decision. As mentioned earlier, if the measurable data are disturbed by noise, the membership functions should be sufficiently wide to reduce the sensitivity to noise. This, raises the issue of specificity of a membership function which affects the robustness of a FLC. This issue is discussed in detail in [29]. (iv) Fuzzy partition of Input -Output space [14]: Fuzzy partition basically determines the number of primary fuzzy sets. The primary fuzzy sets (linguistic terms) usually have a meaning , such as NB (negative big), NS (negative small), PM (positive medium) etc. A typical example is shown in Fig-4 (a,b) . The number of primary fuzzy sets determines the granularity of the control action obtained by FLC . Generally, a heuristic cut and trial method is necessary to find out the optimal fuzzy partitions.

3.2.2

Rule base

The most important feature of the FLC is to generate the fuzzy control rules in an appropriate manner. There are, in general, four modes of generation of fuzzy control rules as reported in [30]. Apart from mode of rule generation, the choice of input (state) variables and output (control) variables, types of fuzzy control rules and justification behind the control rules are are the salient features of the rule base of a FLC.

213

*- trror input 10)

lb)

* - d«fuz7ifi*() outpirt of FLC

Fij 4 : Portioning of tht input-output spoc* .

i) Generation of fuzzy control rules:

(a) Experience and Knowledge of an expert. The fuzzy control rules can be generated in two approaches l) introspective verbalization of human expertise (see Table V) 2) interrogation of experienced experts or operators using an organized questionnaire. The fuzzy control rules provide a convenient way to express the domain specific knowledge through fuzzy if then rules. (b) The decision of an operator: The dimension and the dynamics of the real system are so huge and complex that most of the time it becomes almost impossible to generate the model (in terms of the state space equation or Transfer function). Thus

214 the conventional text book approach to design a model-based controller fails.

But u n d e r such circumstances the skilled h u m a n o p e r a t o r s can

still control such systems successfully without hearing any quantitative models (in t e r m s of quantitative differential equations).

In fact, a hu-

m a n operator generates consciously or subconsciously - a set of fuzzy if then rules to control the system whose qualitative description (in t e r m s of qualitative physics [31]) are built in the mind of the o p e r a t o r . Hence, as pointed out by Sugeno, to a u t o m a t e such processes, it is natural t o express the operator's rules as fuzzy if-then rules using linguistic variables. (c) Qualitative

modelling

of a

system:

T h e design of a FLC based on the above m e t h o d s works well if an e x p e r t / o p e r a t o r plays an important role in controlling the process. But if we want a design independent of of an e x p e r t / o p e r a t o r and if we want to have a design based on the deep analysis of the system dynamics then we have to look for a model of the system. T h e linguistic description of the dynamic characteristics of a controlled system may be viewed as a qualitative fuzzy model of the process. Based on this model we can generate a set of fuzzy control rules for a t t a i n i n g optimal performance of a dynamic system. T h e set of control rules form the rule bcise of a F L C . Although this approach is more rigorous b u t provides a more t r a c t a b l e s t r u c t u r e to deal, theoretically, with the F L C . (d) Self

organization:

T h e scheme of self-organizing control (SOC) has a hierarchical s t r u c t u r e which consists of two rule bases. T h e first one is the general rule base of

215 a F L C . T h e second one is constructed by "meta-rules" which can modify the general rule bcised depending upon the desired overall performance of t h e system. So far, many significant contributions have been m a d e in this direction [32](33l[34l[35[36l[37][38l.

ii) Choice of variables of FLC:

The choice of linguistic variables and their membership function play a significant role in the linguistic structures of a FLC. The linguistic variables in a FLC are the state, state error, state error derivative, state error integral etc.

iii) Types of Fuzzy control rules:

Generally, there are two types of fuzzy control rules, a) state evaluation type b) object evaluation type.

(a) State evaluation type: Most FLC's have state evaluation type fuzzy control rules which can be described as follows;

Ri : ifxisAi

andyisBithenZisCi

Ri : ifxisAi

andyisB2thenZisC2

Rn '•

ifxisAn....andyisBnthenZisCn

where x, y and z are linguistic variables representing the system's state variables and the control variables, A,, Bi and C, are the linguistic values

216

of the linguistic variables a:,..., y, and z in the universe of discourse U, ...V and W respectively, i = 1, ...n. In a more general form, the consequent can be represented as a function of the system's state variables; Ri : ifxisAi...andyisBithenZ

=

fi{x,..y)

The state evaluation type fuzzy control rules evaluate the process state (e.g. state, state error, state derivative, state integral) at time instant t and compute a fuzzy control action at time instant < as a function of (x, ..J/) and the control rules in the rule set. (b) Object evaluation type: The object evaluation type control rules satisfy the states and objectives. A typical example of such a control rule is as follows; Ri : if(uisCi

-^ (xisAiandyisBi))then

uis C^

A control command u takes a crisp set a^ a value and x, y are performance indices for the evaluation of the i — th rule, taking values such zs "good" or "bad". The most likely control action is selected through predicting the results (x,y) corresponding to every control command C;. The above rule can be interpreted as follows; if the performance index x is Ai and index y is 5 , when a control command u is selected to C, then this rule is selected and the control command C, is taken to be the output of the controller [39][40](4l) [42].

iv) Justification of fuzzy control rules:

There are two main ways to justify the generation of control rules ;

217

(a) A heuristic approach when control rules are generated in such a way that the deviation from a desired state can be corrected and the control objective can be achieved. The generation of rules is purely on the quantitative knowledge of the system behavior. Several approaches for adjustment of rule selection have been reported in [43] [44] [45]. (b) A deterministic approach which systematically determine the linguistic structure of the fuzzy control rules that satisfy the control objectives and constraints [46] [47] [48].

3.2.3

Decision making logic

The decision making logic is mainly responsible for fuzzy inference using the method of approximate reasoning as mentioned in Section 2.0.

It has the

following components.

(i) translation of a law of implication to an appropriate fuzzy relational matrix HJI as mentioned in Section 2.1 (ii) application (optional) of the method of redefinition as mentioned in Section 2.2. (iii) change of the insignificant part of a membership function from the exact number zero to a very small positive value. This idea is discussed in [49]. (iv) interpretation of the sentence connective "and", "also", etc.[60]. (v) selection of an appropriate compositional operator, namely ; (a) sup-min operation [20] (b) sup-product operation [50]

218 (c) s u p - b o u n d e d - p r o d u c t operation [51] (d) sup-dreistic-product operation [51] In F L C applications, the sup-min and sup-product compositional operators are the most frequently used. (vi) Inference algorithm which is illustrated in the following using s t a t e evaluation t y p e control laws. Illustration: Rule 1 : AiandBi

=> 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 <„_Bi C Ai oR Proof: Assuming sup-min composition ;

(AioR){v) = sup{min[y4i(u), max [min(ylj(u), Bj(ii)]]} u€U

l<J
>

sup{min[ylj(u),^,(u),i?,(u)]} ueu > 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

•
JJ

bO [sec] Throttle

[ kg/sec] 198 188.8

^ .

pressure

fV. t 1

50 [sec]

Steam flow to H.P. turbine

[cm] 6(.

1 50 [sec]

60 Drum level

Fig 7 : Parameter sensitivity test

232 Once we achieve satisfactory plant responses under different operating conditions we modify the steam pressure/combustion control loop and the drum water control loop of the actual plant (see appendix ) as follows

S t e a m pressure/combustion control circuit: Normally, in real industrial practice, the combustion control system of a steam generating unit (drum type) regulates the throttle steam pressure and C0/02 of the flue gas by manipulating the fuel unit (pulverized coal/coal and oil/coal and gas) and the airflow input to the steam generating unit. In the present problem, we block-decoupled [13] the variables like throttle pressure and flow from the drum water level which disturbs the drum pressure and throttle steam pressure in a nonlinear manner. Once the plant is block-decoupled we apply the following feedback approach as shown in fig.8 for stabilizing the throttle steam pressure. For further detail description of the steam pressure/combustion control loop interested readers are referred to [13|.

Thrott>e pr«ssur» L«ft

Sighl

!PT)

(PT)

govern i-f

D/Al

Mostfr <J«monc) signoL

j ^ n T r »

Pulverizer's feeder-^ Fig.8:St»om pr»»tur»/combusHon

control loops

233

Descriptions

Synibois

©

•

Pressure

Descriptions

Symbols

,'

(ran s m i tier

jfference

[XI

@

Temperotur*

transmider

@

Flow

®

Temperature

(3

Square root

@

Eltctro-pneumotic

CZl

Ht'cjh stgrxii

HuJ

Function

EJ

Low

temperolure

element

extroclor

convertor

selector

ffl

Average

DJ

Transfer

(13

Finol

®

Posit)ori

mdicotor

@

Position

tronsrnitter

<§>

switch

control

element

Setter

-$>•

Auto-n^onuol

selector

IATDI

Anologue to drgitoi

convertor

CPF]

ProportiooQl plus int*gfo( cofltroller

IOTAI

Digital to analogue

convertor

[EDI

Proporttonol plus inrtgrol plus differenfiol controHtr

DO

Multiplier

P j z z y- logic

@

Ratio

reloy

W^

Curve

fitted

ICII —

9#nerotor

signal

cont roll W

Response obtairwd through ;urvefitt»tj controller

controller

Response obtoined througt^ f u i z y logic c o n t r o l l e r

\

234

D r u m water level controller: The function of this control loop is to maintain the drum water level within the prescribed limits. This variable is controlled by modulating the feed water control valves. Two 100% rated full load control valves and two low load control valves are provided. Under low loads, one of the low load valves is taken into service to control the drum water level, the other is acting as standby. When the boiler load exceeds a preset value (25-30% MCR), the control function is changed over through operator's action, to one of the selected full load control valves the other one is serving as standby. The schematic representation of the drum water level control is shown in fig.9.

Feedwo(er level

Drum level

/f]>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 <

error in steam pressure control action (fuel injection)

Table VII: DATA

ei

fi

ei

«i

ei

"1

0.01

0.1

0.1

0.15

0.2

0.2

0.35

0.35

0.5

0.45

0.75

0.5

1.0

0.75

1,25

1.0

1.5

1.5

1.75

1.75

2.0

2.0

3.0

3.0

3.5

3.5

5.0

5.0

7.0

7.0

8.0

8.0

10.0

10.0

12.0

12.0

13.0

13.0

15.0

15.0

17.0

18.0

20.0

20.0

22.0

22.0

25.0

24.0

28.0

26.0

30.0

28.0

33.0

29.0

35.0

30.0

242

Table VIII: FUZZY MODEL If 63

= SMALL then V3

= SMALL

also if 63

= more or less MEDIUM then V3

= MEDIUM

also if 63

= MEDIUM then V3

= very MEDIUM

also if 63

= BIG then V3

= BIG

also if 63

= very BIG then V3

= very BIG

63

< error in drum level

V3

< control action ( opening of control valve)

Table IX: DATA

63

V3

63

W3

63

V3

0.01

0.45

0.2

0.47

0.5

0.5

0.6

0.55

0.7

0.6

0.8

0.7

1.0

0.75

1.25

0.8

1.5

0.85

1.75

0.9

2.0

1.0

2.25

1.25

2.5

1.5

3.0

1.75

3.25

2.0

3.5

2.25

4.0

2.50

4.5

2.75

5.0

3.0

5.5

3.5

6.0

4.0

6.5

4.5

7.0

5.0

7.5

5.5

8.0

6.0

8.5

7.0

9.0

7.5

9.5

8.0

10.0

9,0

243

Table X: MEMBERSHIP FUNCTION FOR ej:

Univ

NULL

very

SMALL

SMALL

more or less

MEDIUM

MEDIUM

more or

BIG

less BIG

very BIG

0.01

1.0

0

0

0

0

0

0

0

0.1

.75

.2

0

0

0

0

0

0

0.2

.5

.5

.3

0

0

0

0

0

0

0

0

0.35

.3

1.0

.5

0

0

0.5

.1

.5

1.0

.2

0

0

0

0

0.75

0

.3

.5

.5

0

0

0

0

1.0

0

0

.3

1.0

0

0

0

0

1.25

0

0

0

.75

0

0

0

0

1.5

0

0

0

.6

0

0

0

0

1.75

0

0

0

.5

.3

0

0

0

2.0

0

0

0

.3

.5

0

0

0

3.0

0

0

0

.2

.75

0

0

0

0

0

0

3.5

0

0

0

0

1.0

5.0

0

0

0

0

.75

0

0

0

7.0

0

0

0

0

.45

0

0

0

8.0

0

0

0

0

.3

.4

0

0

10.

0

0

0

0

.25

.5

0

0

12.

0

0

0

0

.1

1.0

0

0

13

0

0

0

0

0

.75

.2

0

15

0

0

0

0

0

.4

.3

0

17

0

0

0

0

0

.3

.5

.1

20

0

0

0

0

0

.25

1.0

.2

22

0

0

0

0

0

.1

.75

.25

25.

0

0

0

0

0

0

.5

.3

28.

0

0

0

0

0

0

.25

.5

30.

0

0

0

0

0

0

.1

.65

33

0

0

0

0

0

0

0

.8

35

0

0

0

0

0

0

0

1.0

244

Table XI: MEMBERSHIP FUNCTION FOR vn

Univ

very

SMALL

SMALL

more or less

MEDIUM

MEDIUM

very

more or

MEDIUM

less BIG

BIG

very BIG

0.1

1.0

.1

0

0

0

0

0

0

0.15

.5

.5

0

0

0

0

0

0

0.2

.1

1.0

.1

0

0

0

0

0

0.35

0

.4

.5

0

0

0

0

0

0.45

0

.1

.7

.1

0

0

0

0

0.5

0

0

1.0

.3

0

0

0

0

0.75

0

0

.6

.5

0

0

0

0

1.0

0

0

.4

.7

0

0

0

0

1.5

0

0

.1

1.0

0

0

0

0

1.75

0

0

0

.8

0

0

0

0

2.0

0

0

0

.7

.1

0

0

0

3.0

0

0

0

.4

.2

0

0

0

3.5

0

0

0

.3

.4

0

0

0

5.0

0

0

0

.1

.5

0

0

0

7.0

0

0

0

0

.7

0

0

0

8.0

0

0

0

0

1.0

0

0

0

0

.6

.1

0

0

10.

0

0

0

12.

0

0

0

0

.5

.25

0

0

13

0

0

0

0

.3

.3

.1

0

15

0

0

0

0

.25

.5

.3

0

18

0

0

0

0

.1

1.0

.45

0

20

0

0

0

0

0

.6

.6

0

22

0

0

0

0

0

.4

.7

.2

24.

0

0

0

0

0

.3

1.0

.4

26.

0

0

0

0

0

.1

.7

.6

28.

0

0

0

0

0

0

.5

.8

29

0

0

0

0

0

0

.3

.9

30

0

0

0

0

0

0

.2

1.0

245

Table XII: MEMBERSHIP FUNCTION FOR eg:

Univ

SMALL

more or less MEDIUM

MEDIUM

BIG

very BIG

0.01

1.0

0

0

0

0

0.2

.7

.7

0

0

0

0.5

.6

.4

0

0

0

0.6

.3

.5

0

0

0

0.7

.1

.7

.2

0

0

0.8

0

1.0

.3

0

0

1.0

0

.5

.45

0

0

1.25

0

.45

.5

0

0

1.5

0

.3

.7

0

0

1.75

0

.2

.9

0

0

2.0

0

0

1.0

.1

0

2.25

0

0

.9

.2

0

2.5

0

0

.7

.3

0

3.0

0

0

.5

.4

0

3.25

0

0

.4

.5

0

3.5

0

0

.3

.7

0

4.0

0

0

.2

.8

0

4.5

0

0

0

.9

0

5.0

0

0

0

1.0

0

5.5

0

0

0

.9

0

6.0

0

0

0

.8

.1

6.5

0

0

0

.7

.2

7.0

0

0

0

.5

.4

7.5

0

0

0

.4

.5

8.0

0

0

0

.3

.6

8.5

0

0

0

0

.7

9.0

0

0

0

0

.8

9.5

0

0

0

0

.9

10.

0

0

0

0

1.0

246

Table XIII: MEMBERSHIP FUNCTION FOR V3:

Univ

SMALL

MEDIUM

very MEDIUM

BIG

very BIG

.45

1.0

0

0

0

0

.47

.8

.1

0

0

0

.5

.6

.6

0

0

0

.55

.3

.7

0

0

0

.6

.1

,8

0

0

0

.7

0

.9

0

0

0

.75

0

1.0

0

0

0

.8

0

.9

0

0

0

.85

0

.8

0

0

0

.9

0

.7

.2

0

0

1.0

0

.6

,3

0

0

1.25

0

.5

,5

0

0

1.5

0

,4

,6

0

0

1,75

0

,3

,7

0

0

2.0

0

.2

,8

0

0

2.25

0

,1

1,0

0

0

2.50

0

0

.9

0

0

2.75

0

0

.8

.2

0

3,0

0

0

.7

.3

0

3,5

0

0

.5

.4

.2

4,0

0

0

.4

.6

,3

4,5

0

0

.3

.7

,4

5,0

0

0

.2

.8

.5

5.5

0

0

0

.9

.6

6.0

0

0

0

1,0

.65

7.0

0

0

0

.7

.7

7,5

0

0

0

.5

.8

8,0

0

0

0

,3

.85

9.0

0

0

0

,1

1.0

247

7.

Critical appreciation on t h e design of

FLC T h e successful design of a F L C , in a very complex situation, has been claimed by many researchers. But it has been found after a t h o r o u g h investigation t h a t the sole motivation of the design of a fuzzy logic controller to control processes is to incorporate the knowledge of the experienced process operator t h r o u g h some linguistic s t a t e m e n t s which involve the inexact a n d / o r imprecise descriptions of the situations to be controlled. Since the successive input to a fuzzy logic controller is the error between t h e desired value and the measured value which is evidently non-fuzzy in n a t u r e and, therefore, everytime it is necessary to convert the non-fuzzy information to a fuzzy one using some s t a n d a r d technique of fuzzification for further processing by the inference module. Now the fuzzy control algorithms positional

rule of inference

depends

on fuzzy

lattice.

Moreover,

first

defuzzified

injected necessary

which has many

drawbacks

logic, an algebraic system the ultimate

fuzzy output

by some standard

to a plant/system

design of process

of a fuzzy

technique

of approximate

complemented

of defuzziflcation Hence,

which

logic controller and

it is not

reasoning,

is then

absolutely

directly, for the

controllers.

our basic aim is to present a design technique

not suffer from the existing drawback of the compositional as shown in [ISJ and which replaces the idea of with the technique

and com-

[18] and

that is not a

as control action.

to use the technique

In this section,

art based on fuzzy implication

of curve-fitting

which

rule of

does

inference

fuzzification/defuzziflcation

through some points obtained from

the

248 defuzzification demonstrate

of the inexact

that if the final control action

through some assumptions use the technique

at the structural

reasoning.

using curve-fitted

level.

reduces to a non-fuzzy

or approximations

of approximate

result can be achieved

7.1

informations

We value

[4j then there is no need to Rather,

controllers

a very

satisfactory

as well.

Mathematical formulation of the problem

Let X and Y represent two linguistic variables taking values in U = X) u and V = Ylv respectively. We consider a model where we infer from a single fact of the form p «* X ts A'

and a set of rules of the form q •^ if X is AthenYis

B

where A, B, A' are fuzzy subsets of U. V, U respectively, viz. q\ -O- if X is Ai thenYis

Bi = Ri (,say)

also ^2 '^ * / ^ is AithenYis

B-i = R^ (,say)

also •<=> if X isAnthenYisBn

= Rn (,say)

249

where n < Q, A, s are fuzzy subsets of U amd 5 , s are fuzzy subsets of V{i=l,2,...,n). Now we first defuzzify the relations in the premise q. This can be done in two ways. First, we may construct the relational matrices Ri,R2, ••.,Rn independently and then form a unified relational matrix R. Here we can use any translating rule for translating those compound statements to a relational matrix. This matrix is then defuzzified to obtain a collection of pairs of observations distributed over the universe U x V. Such a relation R is defuzzified in the following way : For a particular value of X = u the corresponding values of K = v is that for which the pair (u,v) has maximum possibility in the relation R{u^v). If there be more than one such Y then we may simply take the mean value as a representation of that clciss. The relations Ri, R2.,...,Rn can be formed using many available translating rules [l)[60]. We use two widely used rules for the translation of such compound propositions. They are Zadeh's arithmetic rule, Mamdani's min rule, viz., if X is A then Y is B -* ^T^x,y) = RCU

xV

where fiR{u, v) is either min{l, 1 — ^>i(u) + /ifl(t;)} or

min{nF(u),HB(v)}.

Now if we use Zadeh's rule for the translation then we construct the fuzzy relational matrix R according to R = Ry (~\ R2 f]... f\ i2„ otherwise, we use R = RiDR2D...[JRn

for the same.

Now every relational matrix R of the form discussed so far, when defuzzi-

250 fied defines an elementwise relation between two distinct variables say, current and rotational speed of a motor and gives us informations a b o u t variations of a single variable with a change in the independent variable of t h e system. T h i s relation R, thus formed, is then defuzzified to obtain t h e desired set of points. For n > 1 a second m e t h o d is to defuzzify the fuzzy sets in the antecedent and the consequent p a r t of the conditional s t a t e m e n t s at the very beginning. Now a fuzzy set A can be defuzzified using many different rules [11). We use a simple one : T h e defuzzy value of the fuzzy set A — E ^ ( M A ) ( " ) / ' ^ ) is t h a t value u for which M A I ^ ) is a m a x i m u m in A. If there be more t h a n one such u then we may take the m e a n value as a representation. In this way we obtain pairs of observations if a; = u t h e n y = V from the conditional s t a t e m e n t s if X is A then Y is B. T h u s , in b o t h cases, letting t h e m as points in a two dimensional plane we can always find a best fitted curve t h a t is defined over an extended domain containing U X V. For our present discussion let x and y be two classical variables which takes on values in U and V respectively. Let f : U => V he t h e function generated from those defuzzified points by any s t a n d a r d technique of curve-fitting. Let's now defuzzify the proposition p to obtain x = u T h e n from y = f{x)

we obtain / ( u ) as the desired

inference which can be directly injected to the plant as control action.

7.2

Numerical example

Let us consider the following problem in which we are given three fuzzy propositions of the form if X is A then Y is B

251 where X a n d Y are two linguistic variables defined over U a n d V respectively a n d a n o t h e r fuzzy premise of t h e form Xis A' where A, A' are fuzzy subsets of U and B is a fuzzy subset of V. Let [7=1 + 2 + 3 + 4 +5, V = 6 + 7 +8. Let the three compound assertions be qi O ifX

is 'low' then Y is 'low'

92 ^^ ifX

is ' m e d i u m ' then Y is ' m e d i u m '

qs -O- ifX

is 'high' then Y is 'high'

and in terms of possibility distribution.

qi ifXis{l/l

+ .75/2 + .5/3 + .25/4 + 0/5)thenYis{l/6

+ .6/7 + .3/8)

52 O ifXis{.5/l

+ .75/2 + 1/3 + .75/4 + .5/5)thenYis{.6/6

+ 1/7 + .6/8)

93 <^ ifXis{Q/l

+ .25/2 + .5/3 + .75/4 + l/5)thenYis{.3/6

+ 6/7 + 1/8)

The corresponding relational matrix (using Zadeh's arithmetic rule) will be Y

6

7

8

X

6

7

8

Y

6

7

X

X

1

.6

.3

1

1

1

1

1

1

1

2

1

.85

.55

2

.85

1

.85

2

1

1

3

1

1

.8

3

.6

1

.6

3

.8

1

4

1

1

1

4

.85

1

.85

4

.55

.85

5

1

1

1

5

1

1

1

5

.3

.6

1

Ri,=

Y

setting It =^ Jti n 112

and lettin g

niZs

8

252 fJ-R{x,y) =

mm{niti{x,y),fj.K2(x,y),lJ.R3(x,y)}

we find Y

R =

1

1

.6

.3

2

.85

.85

.55

3

.6

1

.6

4

.55

.85

.85

5

.3

.6

1

Let the fuzzy proposition X is A' induces a possibility distribution (.25/1 + .50/2 + .75/3 + 1/4 + .75/5).

T h e n compositional rule of inference gives the induced possibility distrib u t i o n of Y as (.6/6 + .85/7 + .85/8). Whereas defuzzification of relations RijR^^Rs

gives

iii ^^ if X is 1 then y is 6 ^2 ^^ if X is 3 then y is 7 iia ^ if X is 5 then y is 8

X and y are defined over U and V respectively.

Letting t h e m as point

in a two-dimensional plane we find t h a t the corresponding interpolating

253 polynomial will b e [12] y = 5.5 + 0.51

3

Now the defuzzy value of the proposition XisA'

<=!> TTA: = (.25/1 + .50/2 + .75/3 + 1/4 + .75/5)t5a; = 4 and

from t h e relation (3) for a: = 4 we find y — 7.5, which is the same as the defuzzy value of "Y is (.6/6 + .85/7 + .85/8) ". But these two values of y m a y not always be equal; instead they may be close to each other.

7.3

Implementation of the proposed curve-fitted con-

troller

To measure the effectiveness of the newly proposed curve-fitted controller we test the asymptotic responses of a real closed-loop plant (see a p p e n d i x ) u n d e r different operating conditions stated in section 6.0. We design the following two expressions for curve-fitted controllers of the plant using the d a t a in tables X, XI, XII, XIII. Curve-fitted control action with respect to the closed-loop plant as shown in the appendix , are as follows. vi = 0.0015e^ - 0.1083eJ + 2.7857ei - 0.6363 t;3 = 0.452 - 0.18263 + 0.807e^ - 0.15e^ + O.OOSe^.

We compare the closed-loop plant responses with those obtained t h r o u g h FLC in Figure 5, 6, 7, 8, 9 and 10 and satisfactory results are achieved.

254

Table XIV: PERFORMANCE OF THE THREE CONTROLLERS ON A STEP RESPONSE OF 10 (kg./sec) STEAM FLOW Controller

Rise time

Overshoot

Steady state

(sec.)

(kg./sec.)

error (kg./sec.)

1

1.85

- 0.2

Fuzzy logic

0.5

2

0.195

Curve-fitted

0.56

1.89

0.075

PI

7.4

Comparison of system performances

In tables XIV and XV we summarize the performances of the conventional two-term controller (PI), conventional fuzzy logic controller [13] and the proposed curve-fitted controller. For comparison, we consider two-term controller simply because they are the existing control actions for conventional steam flow and drum water level control loop of the steam generating unit as described in the appendix . From the given tables, it is obvious that the proposed curve-fitted controller is as good as a fuzzy logic controller for an easy implementation of control action for nonlinear plant. The switching line for the bang-bang control or variable structured control of a nonlinear plant is difficult to calculate. Hence the simplicity of the curve-fitted controller and the fuzzy logic controller is an added advantage over the classical approach to controller design. Finally, in Figure 11 we have shown that the steam flow and throttle

255

Table XV: PERFORMANCE OF THE TEREE CONTROLLERS ON A STEP RESPONSE OF 8 (cm.) DRUM WATER LEVEL Controller

Rise time

Overshoot

Steady state

(sec.)

(cm.)

error (cm.)

1.2

1

-0.2

Fuzzy logic

0.35

0.9

0.1

Curve-fitted

0.4

0.5

0.08

PI

pressure characteristic is closely tracked by the newly proposed curvefitted controller and conventional FLC. Hence, we can conclude, based on the results obtained, that the proposed curve-fitted controller is as good as the fuzzy logic controller.

USKg/Sec -

Plont Charactertshc Inference obtained by conventionoi fu72y logic controller

135Kg/S»c ISlKg/Cm'

St»om pressure

175Ks/Cm'

Fig. 11; Trocking ot nonlineor characteristic of the plant.

256

8.

conclusion

In t h e era o f m a t h e m a t i s a t i o n " of control engineering problems, F L C can be seen as lying in the general s t r e a m of expert control, characterised by the a p p r o p r i a t e mechanism for representating vague h u m a n j u d g e m e n t s . But a prominent problem in the design of the FLC is to transfer the expert knowledge into the usable knowledge beise of the F L C . Hence, we need a very sophisticated tool for knowledge acquisition for the design of the F L C . In control engineering, FLCs are extensively studied. Various CEise studies present their applications in various process-control systems previously controlled by conventional control algorithms.

One has to u n d e r s t a n d

clearly the notion of "fuzziness" in process control. Apparently it may be seemed t h a t application of fuzzyset to process control is a forceful imagination of a designer. B u t such an impression is not always t r u e . In control strategy applied by the operator we can recognize some concepts t h a t are evidently fuzzy in n a t u r e , for instance, if observed variable x is "positive m e d i u m " , then change the control variable y by the a m o u n t "negative m e d i u m " . In table XVI we provide some milestones in the development of the FLC (choice of milestones may be subjective). T h e design of F L C based on the fuzzy model of a process is still not well u n d e r s t o o d due to difficulties in modelling the linguistic s t r u c t u r e of a process and obtaining d a t a in industrial process control [60]. In comparison with conventional control theory, at present, the design of F L C is less analytic and needs well-founded procedures for system design. In response to this need recently we have seen some developments

257 Table XVI: M I L E S T O N E S IN T H E D E V E L O P M E N T O F FLO

1972 Zadeh

A rationale for fuzzy control

1973 Zadeh

Linguistic Approach

1974

M a m d a n i
Steam engine control

1976

Rutherford

Analysis of control algorithm

1977

Ostergaard

Heat exchanger & cement kiln control

1980

Tong et.al

Wastewater t r e a t m e n t process

1980

Fukami, Mizumoto

Fuzzy conditional Inference

&: Tanaka 1983

Hirota & Pedrycz

Probabilistic Fuzzy sets (control)

1983

Yasunobu, Miyamoto et.al

Predictive Fuzzy control

1984

Sugeno & Murakami

Parking control of a model car

1984

Ray & D u t t a Majurader

Analysis of Fuzzy logic controller

1985

Kiszka, G u p t a et.al

Fuzzy system stability

1985

Ray & Ghosh &

F L C Stability

D u t t a Majumder 1986

Yamakawa

Fuzzy controller hardware system

1987

Ray

FLC for Multivariable Plant

258

(14][17l[83l [841(85]. The hardest part of designing fuzzy controller is selecting which fuzzy sets best represent the controlled and controlling variables, namely, the "tunning" of the controller. Most controllers are sensitive to the shapes of these membership functions, and as the number of rules multiples, trial-and- error tuning methods becomes less feasible. A recent breakthrough is artificial neural network, which with the aid of back propagation learns the needed membership functions from a set of training examples. The back propagation type carries out supervised learning by leading output failure to match a desired target pattern back to its input as information to its individual nodes. Hideyuki Tagaki and Isao Hayashi researchers with Matsushita's Central Research Laboratory, Osaka, first reported these results at the 1988 International Conference on Fuzzy Logic and Neural Networks, lizuka, Fukuoka.

Subsequently,

Akira Maeda and others of Hitachi's Systems Development Laboratory, Tokyo, have developed a tuning system on this idea and applied it in the development of a controller design previously by trial and error. Recently we have seen some theoretical development for self- organizing control using fuzzy neural networks|86j.

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Appendix Physical description of the plant [13] General description of the plant: The 210 (MWJ thermal power station was designed on the concept of the unit system, where single steam generator supplies steam to a single turbine coupled with an AC Generator. S t e a m generator plant: The steam generator made in the BHEL is of single drum water tube natural circulation type with superheaters and reheaters.

The maxi-

mum continuous evaporation of the steam generator is 680[Te/hr.],

at

I54.15[kg/cm^\ and 540[°C]. The final feed water temperature is 243[°C]. The complete furnace section is of welded wall type, arranged as a gas and pressure-tight envelope.

It has extended side-wall section, where

reheater is also covered with water-cooled section. The circulation system is complete with unheated downcomers, supply and riser piping.

269

Table XVII: STEAM PARAMETERS AT DIFFERENT CONDITIONS : At p = 210IMWI

Top heater

At p = 168[MW1

VWO

CW Temp. 33

out At p = 210

MU - Ol%)ClV

p = 221.5[MW'i

Temp. 33[°C]

[°C\MU - 0(%]

[MW\MU

and back

Back

- 0(%|

CW Temp.

MU-0|%|

Pr.0.1033 ata

Pr.0.1033 ata

33|»Cl

(1)

(2)

(3)

(4)

T/H HPT

627.319

594.733

500.649

666.958

IPT

565.602

588.178

454.393

600.287

inlet

241.0

199.8

230.9

243.9

temperature

\"c\ (1)

(2)

(3)

(4)

Steam Bow

F W econ.

1 0.20(37

0.2122

0.1710

0.2178

2 0.8037

0.8339

0.6581

0.8534

steam

3 2.2511

2.3251

1.8411

2.3770

pressure

4 6.5805

6.7589

5.3921

6.9437

5 16.043

16.597

12.989

16.992

6 36.363

0.0

29.542

38.464

(1)

(2)

(3)

(4)

Extraction

Condenset temp. after ejector 47.4

47.3

45.4

48.0

G.S.C.

47.7

47.7

45.9

4S.4

D.C.

48.6

48.5

46.5

49.2

LPH-1

56.8

57.1

53,7

57.7

LPH-2

90.3

91.0

85.7

91.5

LPH-3

120.5

121.4

U4.9

112.1

Deaerator

160.8

161.9

152.9

163.0

HPH-5

197.9

199.8

189.7

200.2

PIPH6

241.0

199.8

230.9

243.9

270

S c h e m a t i c r e p r e s e n t a t i o n a n d m a t h e m a t i c a l d e s c r i p t i o n of t h e plant : The schematic representation of the system under consideration is shown in fig.Al. The major control variables are fuel input Pp, feedwater input Fw and control valve setting Cy. The output

FP^O vVQter input

Control valve of H, R torbine

Fig l A Schematic representotion of the steam

generoting

system

271

variables are throttle pressure P, steam flow to H.P. turbine S and drum water level L. Thus, a nonlinear multivariable plant having three inputs and three outputs is considered.

The validity of the model is tested

with real industrial data of 210 [MW] unit (BHEL, INDIA). The plant is assumed to be operated at 60% maximum continuous rating (MCR). A step change of ± 10 - 20% around the operating conditions of the plant is injected into the open-loop plant. It is observed that the open-loop plant dynamics shown in fig.A2 agree quite well with real plant responses. The details of the system model is given in [13].

[cm] 68 2 6 A 5 -

64 0267 Drum 'evel • i~ 62 8 2 5 3 - 5 0

2'.5

T = 3000[sec]

SO

175.8 TJ 3000 Drum

pressure

151.30 i l<9 / s e c j U 8 36 SteOm

flow

135,0 8

' -.'i°'o' change in control

T = 300n(sec]

valve

opening

272

73.96r

^^^•-^

6*00

Drum le»el T--3000l»c)

H«9/s»cJ 135-00 ^ ' ^ - - - _ _ _ _ S < . o m fto* 133-85

T = 3000ls»c)

[kg/cm'] 175.7968, 175.8 k \\..___D™m 172,8231 3

pressuro 1-3000[s«)

lb) 10-20CV.] chonge in f»»Jwot»r input

I c '"J

Drum level T= 3000 [sec]

51 S6.5*

kg/cm ] 209.3117 6 8

T-SOOOlsecj

[ kg/sec] 11.7.30 Sfeom

flow

135.0

T = 3000 [sec]

Ic) 10-20r/«]

change

F i g . 2 A : Op*n-toop

in fuel

plont

input

dynomUs.

273

The set of first order nonlinear differential equations which represent the system's dynamics in time-domain is given by dPIdt

=

{-0.001935.P. + 0.014525Pf - 0.000736Fw

dS/dt

=

(lOCv.pi/2-0.7857165)

dLjdt

=

(0.000863Fiv + 0 . 0 0 2 P F + 0.463Cv - O.OOOOOeP^

+0.00121i + 0.000176r£') {A\)

-0.00914L - 0.0000821^ - 0.0073285)

In the above description of the model, the variable 'TE', which represent the feed water input temperature, may be reasonably assumed to be constant (no heater [high pressure or low pressure] is tripped during operation). However, if any fluctuation in the variable 'TE' occurs due to either improper heat transfer in the economizer section of the boiler, through which the input feedwater finally passes, or sudden tripping of any heater etc., that fluctuation of 'TE' becomes very unpredictable in nature. Hence, for simplicity of the decoupling controllers structure, which is based on the mathematical description of the nonlinear multivariable plant/system, the variable 'TE' of the model (Al) is set to zero. Thus we get the following simplified mathematical description : ii i2 X3

-0.00193i2i?^^^ + 0.0012113 =

-0.785716i2 - 0 . 0 0 0 0 0 6 1 ? - 0.0073283;^ - 0 . 0 0 9 1 4 i 3 - 0 . 0 0 0 0 0 2 i § 0.014524

0.0

-0.000736

"1

0.0

lO.Oa:^*

0.0

"2

0.002

0.463

0.00863

U3

(A2)

Yz

1

0

0

Xi

0

1 0

X2

0

0

X3

1

The state variables (P,S,L) = (xi, 12)2:3) and control inputs (Pp, Cv,Fw)

274 (UIJUJJWS), respectively.

The basic design task is divided into two parts : (i) Deriving the nonlinear state variable feedback law for the block decoupling of the plant [13] (ii) Generation of the FLC/CFC for the block- decoupled subsystems. The three outputs of the plant are steam flow to H.P. turbine, throttle pressure and drum water level. Among these outputs, two of them (steam flow to H.P. turbine and throttle pressure) will be controlled together and the remaining one (drum water level) will be controlled separately. A careful inspection reveals that the second output (i.e., the steam flow X2) of the plant represented by (A2) does not depend directly on drum water level (xs). Hence, in the present problem, for the sake of block-decoupling, the first output will be straightway decoupled from the third. Thus, the first and the second outputs will be decoupled from the third, whereas the first and the second outputs will remain interacting within themselves. Here, the advantages of a particular mathematical description (A2) of the given steam generating unit will be exploited for block-decoupling [13]. The final closed loop plant is shown in fig.(A3).

- 1

Blrjck-decoupled inner loop

ir ^ '-^

^

FLC/CFC

—n G*(x) ^

^ ^

y^

11

Nonlinear

^

eqOQtions

^ L Blok decoupled outputs

r i x ) * <J

I

I

I

I

Outer-loop

cU)

feedbock

f-ig 3 A Schemotic representation of the closed-loop

system

This page intentionally blank

275

Acknowledgment The author is thankful to Maya Dey, Joydeb Gupta, Somnath Mazumder and Probal Sengupta for preparing the manuscript. Author is also thankful to Swapan Raha from whom author has learned many new aspects of fuzzy logic and its applications.

CHAPTER 10

FUZZY CONTROLLER DESIGN: A SLIDING MODE APPROACH Chung-Chun Kung and Sinn-Cheng Lin Department of Electrical Engineering Tatung Inst, of Technology 40 Chungshan North Road, 3rd Sec, Taipei, Taiwan, R.O.C.

Abstract A fuzzy-sliding mode controller, which is designed by the techniques of the fuzzy logic controller and the sliding mode controller (or called variable structure control), is proposed in this work. Like the sliding mode of the sliding mode control system, the fuzzysliding mode control system has a fuzzy-sliding mode. The reason for calling "fuzzysliding mode" is that the sliding surface in the proposed scheme is a fuzzy set rather than a crisp set found in the conventional sliding mode control system. In the design of the fuzzysliding mode controller, one can easily determine the membership function, observe the fuzzy rules and predict the controlled system behavior. Furthermore, the number of inference rules, which is an exponential function of the number of system state variables in a conventional fuzzy logic controller, is reduced to a linear function of the number of system state variables in the fuzzy-sliding mode control system. Simulation results show that the proposed scheme has the following advantages: 1. The dynamics behavior of the controlled system can be approximately dominated by a fiizzified sliding surface. 2. Fuzzification of the sliding surface will not only increase the robustness to the system nonidealities but also decrease the chattering. 3. The fuzzy-sliding mode controller can control most of the complex ill-defined systems without knowing their mathematical models.

277

S.G. Tzafestas and A.N. Venetsanopoulos (eds.) Fuzzy Reasoning in Information, Decision and Control Systems, 277-306. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.

278

1. Introduction The fuzzy logic control has become one of the most important areas of research in the applications of the fuzzy set theory [l]-[4] since about twenty years ago, Mandani and his co-workers [8]-[10] have successfully applied the simple_/Hzzy logic controller (FLC) to several industrial processes such as the steam engine, warm water plant, stirred tank temperature system, etc. During recent years, the literature fuzzy logic control has been growing rapidly. Most of the historically important milestones in both the developtnent and application of fuzzy logic control have been summarized by Lee [5][6]. By converting the linguistic control strategy into automatic control strategy without using the mathematical model of controlled systems, the FLC can be used to deal with complex ill-defined systems. However, the work of FLC design is time-consuming and the response trajectory of the controlled system is unpredictable because fuzzy control rules are experience oriented and the suitable membership function should be selected by the trial and error procedure. To overcome these disadvantages, we will introduce the sliding mode of the sliding mode controller (SMC) to the FLC. We will call it the fuzzy-sliding mode controller (FSMC). The conventional sliding mode control methodology has been developed in the literature [15]-[17]. By choosing a suitable sliding surface X = { X I cj(x) = 0 } and applying discontinuous control action across the surface, the resulting state trajectory of the controlled system can be transferred onto the sliding surface and then slide along the surface. There are two phases in the sliding mode control system. One is called the "hitting" phase and the other is called the "sliding" phase. In the hitting phase, a hitting control law will be applied to transfer the representation point everywhere of the state space onto the sliding surface. As soon as the representation point hits the surface, the controller turns the sliding phase on, and applies an equivalent control law to keep the states lying on the sliding surface. In an ideal situation, when the state trajectory stays on the sliding surface, the control system then has perfect insensitivity with respect to the external disturbances as well as the plant parameter variations. Furthermore, even if the system is perturbed such that the representation point is out of the sliding surface, the control system turns on the hitting phase again, and reacts the hitting control law to bring the states toward the sliding surface once more. Therefore, the sliding mode control imposes certain constrains on system dynamics; in other words, one can predetermine a sliding surface to dominate the dynamic behavior of the controlled system. However, the methodology of sliding mode control system has also some drawbacks as follows: 1. The state trajectory in the hitting phase is usually sensitive to system uncertainty. This difficulty can be alleviated by using high gain feedback to save hitting time [21] [22].

279

But there still exist some problems associated with high-gain feedback such as extreme sensitivity to unmodclled dynamics, actuator saturation, etc. [18]. 2. Due to various nonidealities such as hysteresis of switching operation, time delay of the control system, and sampling of digital implementation, the state trajectory of a realized sliding mode control system does chatter rather than slide along the sliding surface [15]. This undesirable chattering phenomenon will cause high-frequency unmodelled dynamics in the controlled system. In the course of designing FSMC, we first adopt the methodology of SMC to select both the membership functions and the fuzzy inference rules, and then fuzzily the sliding surface and get the feedback control gain from the fuzzy inference rule base. Because the FSMC is composed of FLC and a SMC, it might have the advantages of both.

2. Fuzzy Logic Controller and Sliding IMode Controller 2.1 Fuzzy Logic Controller The simple FLC based on fuzzy set theory [l]-[4], originally conceived by Assilian and Mamdani [11], is used to regulate the output of a process around a given set-point. The basic configuration of the simple FLC comprises three principal components: ajuzzi/ier, a rule base with inference engine, and a defuzzifier as shown in Figure 1. We briefly describe the functions of them as follows: A. Fuzzifier The main function of the fuzzifier is to convert crisp data into fuzzy sets, i.e. A'=F(x)=F{XaXGX) B' = F(y)=Fiy,xGY)

(1) (2)

where A' and B' ate two fuzzy sets, and F(.) represents a fuzzification operation [2], Xg and _Vo are the crisp input values from the process, GXand GY arc the scaling factors of x„ and yg such that x e (7 and ysV. B. Rule Base and Inference Engine In general, an FLC consists of a set of linguistic rules which define the individual control situation. For instance, the control rules of two-input-one-output FLC have the form: R^: If X is Aj andy is Bj Then zisC^,j=

1,2,..., /

(3)

280

where X and y arc linguistic variables representing two process state variables, and z is a linguistic control variable; A., B. and C, withy = 1, 2,..., /, are linguistic values of x, y, and z in the universes of discourse U, V, and W, respectively. The fuzzy control rule is impleniented as a fuzzy relation in C/ X F x W and is defined as follows

R.=

(A^AB,)^q

(4)

In this work, we adopt the widely used Mamdani's mini-operation fuzzy implication function [9], and hence the membership function oiR^ is given by \iiR^{u,v, w) = min{\iAj{u), \i.Bj{v),ViCj{w)]

(5)

Thus the overall relation, R, of equation (3) can be implemented by using the union operation to link all rules together; that is R= RjU R^u ...\J R, = \J Rj

(6)

Consider now, i{A' and 5 ' denote the linguistic values of inputs x and y, respectively, then the linguistic value of output z can be obtained by using the max-min compositional rule of inference

C'^{A'nB')°R

(7)

C. Defuzzifier Because the control signal of a process must be a crisp value signal rather than a fuzzy one; the linguistic value of control output, C, which is obtained from inference engine, must be converted into a crisp value, z^, symbolically

z,= D{C')

(8)

where D(.) denotes any function of dcfuzzification. The one which is widely used and will be adopted in the defuzzifier of FSMC is the center of area (COA) method [13]. GX Fuzzifxer

Rule Base and Inference Engine

GZ

c;

Defuzzifier

4 ^ .

GY

Figure 1 The configuration of a simple FLC

281

To speed up the inference and to save memory one can preconstruct a decision Junction (or, approximately, a look-up table) based on Eqs. (1), (2), (6), (7) and (8) for the FLC. For example, consider an FLC with error e and change in error Ae as two input variables, and change in control Au as one output variable, then one can formulate u{t + At) = u{t) + AH and Au = D{{F{exGE)r\F{AexGQ\ = M{exGE,

oR] x GU

AexGQxGU

(9)

where M{., .) denotes such a decision function.

2.2 Sliding Mode Controller The conventional sliding mode control is also called the variable structure control, which has been developed essentially in the Soviet Union [15]-[17]. Consider a system to be controlled which is described as follows: i = f(x)-H b(x)M

(10)

where x = [Xj x^ ... x„]^ is an /i-dimentional vector that denotes the state of the system, f is an n-dimensional vector, M is a scalar control input and b is an «-dimensional control gain vector. A. The sliding phase Let us define the sliding surface in the state space R" as 2:={x|o(x) = 0}

(11)

where o(x) is an arbitrary function of x which will dominate the dynamic behavior of the control system and have to be properly chosen. From Eq. (10), we see that c r - ^ i = g ' ( x ) [ f ( x ) + b(x)H]

(12)

in which g^(x) is the gradient vector function of a(x). Without loss of generality, we assume that g^(x)b(x) > 0 and is bounded. In the design of the sliding mode control system, we must first find the equivalent control law, u^^, which will keep the state of the controlled system staying on the sliding surface whenever X6 Z, that is

282

o\.^^=0

(13)

Substituting Eq. (12) into Eq. (13), we get g\x)[f(x) + b ( x ) « J = 0

(14)

".,= -(g'b)-'g^f

(15)

Hence

Therefore, under the sliding mode, the equivalent closed loop system can be obtained by substituting Eq. (15) into the original system of Eq. (10), as follows: X = {I-b(x)[gV)b(x)]-'g^(x)}f(x) = f^x)

(16a)

O(x) = 0

(16b)

and

From Eq. (16), we see that the gradient of the sliding surface, g^x), will govern the dynamic behavior of the control system in the sliding mode. Thus one can expect to obtain good performance from the control system by suitable design of the sliding surface. B. The hitting phase In the hitting phase, where xe R°-Z, a hitting control law will be found to transfer the state anywhere to reach the sliding surface. Let us define a Lyapunov function as follows: V = |o'(x)

(17)

It is shown in [19] that if there exists a positive constant ri, such that V<-Tl|a(x)|,

V x e R°-Z

(18)

where r||CT| is a function of class K [23], then the state trajectory will hit the sliding surface, Z, within a finite time /^ defined by ^ |o(x(0))| /, < ^ - ^

(19)

The above results show a global sliding condition and guarantee that all initial state lying off Z will be attracted to Z. From Eqs. (12), (17) and (18), we see that the sliding condition is satisfied if

283

y=aa=a

g^(f + bw) < -ri|cr|

(20)

or, equivalently

ag^ (f + bM) < -TiCT sgn{a)

(21)

where

sgn(a) = \ ^ ^ ^

'' ' i * ' ^ ^

' - 1 , if

(22) CT<0

^

Define a(x) = ri(g^(x)b(x))"', then Eq. (21) is satisfied if « = «., + «j

(23)

where Mj.^ is given by (15) and

u^ > a(x), with o(x) < 0 u' < -a(x), withCT(x)> 0

(24)

Therefore, we can choose « = M., - k(x) sgnic)

(25)

in which k: R " ^ R " satisfies k(x) > a(x) for all xe R". In the ideal sliding mode control, as soon as the state trajectories of the system hit the sliding surface, the state trajectories slide along the surface. However, because of nonidealities of the real systems, a realized sliding mode control system may have chattering along the sliding surface. To alleviate the chattering phenomenon, many strategies have been proposed. For instance, Slotine and Sastry [18] suggested a quasi-sliding mode controller which replaces the term sgn{o) in Eq. (25) by sat{o/e), i.e.,

.,«(a)..aW = {f; l^^^ll

(26)

This leads to tracking within a guaranteed precision e while allowing the alleviation of the chattering phenomenon.

284

3. Fuzzy-Sliding Mode Controller 3.1 Basic Considerations To date, the fuzzy logic control and the sliding mode control have been two different research topics. The basic idea of combining these two methodologies together and formulating a scheme of fuzzy-sliding mode control is that the appearance of nonidcalitics such that the sliding surface of the sliding mode control system is no longer crisp. In other words, it is possible that a state, say x^, which belongs to Z in the ideal sliding mode but does not belong to the same set as nonidealities appear. Hence, the sliding surface turns into a fuzzy set rather than a crisp one in the real situation, and the fuzziness of such a set depends on the amount of nonidealities. Therefore, we can use the fuzzification operator to fuzzily a crisp sliding surface and then obtain a fuzzy-sliding surface. Then, we use the fuzzified version of the crisp variable a as a linguistic input variable of the FSMC. To handle more information, we also fuzzily the derivative of a as another linguistic input variable. Hence, an FSMC uses the linguistic variables of a and d as inputs instead of error and change in error in the conventional PLC. In order to develop the FSMC, we rewrite the FLC method and the SMC scheme as follows: A. Fuzzy Logic Control Law u{t + At) = u(t)+ Au

(27)

Au = M(ex.GE, AexGQ xGU = D{[F(exGE)nF(^AexGQ\oR}xGU

(28)

B. Sliding Mode Control Law

« = «., + u,

(29)

u, = -k(x) sgn(a)

(30)

with

where k: R - ^ R " satisfies k(x) > a(x) for all xe R", and a(x) = r|(g^(x)b(x))' . 3.2 Fuzzifications of Sliding Surface and Its Derivative In the approach of designing FSMC, we will use the fuzzy logic control law of Eq. (27) to replace the sliding mode control law of Eq. (29); namely, the fuzzy sliding mode control law will be formulated as

285 u{t + At) = u(t) + Ujf

(31)

where Ujf is obtained by fuzzy relation and what we wish is that E{u(f)} —> M^^ as ? —> <» (£{•} denotes statistical expectation). To establish such a fuzzy-sliding mode control law, the designer must keep in mind some physical meaning and geometry interpretation of the sliding mode regime. We illustrate these concepts by viewing the dynamics of a linear system. Referring to Eqs. (10), (11), suppose that f(x) = Ax, b(x) = b,CT(x)= c^x, and g^(x) = c^, then, the global sliding condition for such a system is given by (32)

acKO and the sliding mode control law can be chosen by « = "., + " . = - ( k . / + k / ) x

(33)

in which k^^, the equivalent feedback gain vector, and kj, the discontinuous feedback gain vector, are respectively given by k,/ = (cVc"A

(34)

and k'

ko > 0, 0, -^0 < 0,

if csgniXi) > 0 if Gsgn(Xi) = 0 if C!Sgn(,Xi] < 0

(35)

where kj is the /-th element of the vector k^. Substituting Eqs. (33), (34) and (35) into the original system of Eq. (10), one can obtain i=(A-bkJ)x-bk/x

(36)

<j= c^ X = c'r(A - bkJ,) X - c'^bkj x = - c^'bk/x

(37)

and

From the viewpoint of geometry, any state x can be represented as a point in the state space. Thus, the value |CT(X)| can be interpreted as the distance between the representation point X and the sliding surface Z. Consequently, in the physical sense, d can be viewed as the velocity of the representation point tends toward the switching surface. From Eq. (37) one can obtain

c =

'bZkl

= C^bko Hxi

(38)

286

Hence, the magnitude of discontinuous feedback gain, k^, governs the convergent rate of O. Kkg is increased, the speed of representation point that moves toward the surface accelerates and the hitting time is decreased. It also has been shown that a sufTiciently large gain can guarantee the existence conditions of the sliding mode. In practice, however, as the switching operations do not occur as soon as the sign ofCTchanges, the larger gains cause the larger trajectory chattering. This is due to the switching criteria in the conventional sliding regime based on a sign function, which is a crisp version and has a sharp edge. In other words, the sliding surface in the conventional sliding mode control is a fuzzy singleton with membership equal to 1 as o = 0 and equal to 0 as (J ;* 0. Now, let us define the following fuzzy sets: CT.p : a is positive a.z : o is zero CT.n : a is negative such that \i^^B [0, 1], H„^G [0,1] and ^l„„e [0, 1]. It is obvious that in an ideal sliding mode, the above fuzzy sets degenerate into crisp sets, i.e., CT.p={K|a(x)>0}=5:P

(39a)

a.z={x|CT(x) = 0} = 2:

(39b)

a n = { x | a ( x ) < 0 } = ZN

(39c)

and their membership functions are I 1, / /

CT>0

,.„ ,

1, if 0 = 0 0, if CT^tO

(40b)

, 1, i/ o < 0

,.„ ,

^''"io; ,/a>o

('«^>

Similarly, in the quasi-sliding nrode proposed by Slotine and Sastry [18], the switching surface becomes a boundary layer, which is also a crisp set, thus CT.p= { x | a ( x ) > e }

(41a)

287

az = {x||a(x)|<e}

(41b)

a.n = {x I a(x)<-e}

(41c)

and their membership functions are

_ J 1, if a>e

]i.(5.p —

(42a)

0, if CT<e

(42b)

\i.a.,.

1, if a < - e 0, if CT>-e

(42c)

In real situations, the switching surface is a fuzzy set rather than a crisp one. So, the first step in the design of FSMC is to fuzzify the sliding surface; that is a.p=F(XP)

(43a)

a.z = F(2:)

(43b)

a.n = F(EN)

(43c)

In the following discussions, we will set

^(j.p(cr) =

1, oils. 0,

if if if

0 > / j

0 < a < /j

(44a)

a <0

^ o ^ ( o ) ••

if 0 < a < /s I ^ alls. if - / j < a < 0 if 0, |a|>/.

|Xo.«(CT) = •

-alls. 1, 0,

l-CT//„

if - /, 0 if

(44b)

(44c)

288 Similarly, we can fuzzify Aa(x(t)) = C(x(t)) - a(x{t-At)) = oAt , change a, in the same way, such that I, ^4o.p(Aa) =

ACT//,,

0'

if if if

' l-Aa/lc, ;if ^ia^(Aa) = < l+Ao/Zc, if if

n&ani^o) ='

' -Aa/lc, 1, 0'

ACT>/C

0 < ACT < /,

(45a)

ACT<0

0 <

ACT

< /c

-/C
(45b)

1 ACT] > Ic

if - /c < ACT < 0 if ACT<-/<: if ACT>0

(45c)

Now, wc are ready to develop the scheme of fuzzy-sliding mode control.

3.3 The Rule Bases From preceding discussion, we notice that a and ACT can be interpreted as the distance from the representation point to the sliding surface and as the velocity of representation point tends toward the sliding surface, respectively. Furthermore, we have indicated that larger feedback gains cause faster convergence but larger trajectory chattering. To attenuate the chattering, the feedback gains must be chosen appropriately. Therefore, in designing the FSMC, we will use a fuzzy relation to obtain a nonlinear state feedback control gains based on the specification ofCTand ACT. To do this, let us select two groups of variables as the input variables of the FSMC M, = osgn(x),

/ = 1, 2, ..., n

(46a)

V, = Aasgn{x),

I = 1, 2

(46b)

n

Then we transfer these two groups of variables to their corresponding universes of discourse by separately multiplying two groups of scaling factors, i.e., 5, = M, X GS, ,

.y,e tT,,

cs, = V, X GCS. , cj,e V-,

i = 1, 2,..., n i = 1,2,..., n

(47a) (47b)

289

Consequently, the fuzzy-sliding mode control law of Eq. (31) can be formulated as follows: uit + At) = u(0 + Ujj. where

n

«rf/ = -krfy-x = - E t^j-Xi

(48a)

r=l

and k',f = M,{s,,cs;)xGK,

(48b)

Thus, for an n-state system, we should establish n sub-rule bases to form an overall rule base. Each of them has two inputs s. and cs. and one output kj. If we define S-, CS. and KD. as the linguistic variables corresponding to 5,, cs- and k^^ respectively, the fuzzy sets A^j, Bjj and Q ,j = 1, 2, ..., /„ as the linguistic values of 5, , CS^ and KD. respectively, then the rule base may have the following form: Rf (Rule Base i, RB-,

i=\,2,...,n)

R.,:

If S,.A,,andCS,.B.,

Then KD,C,^

R-f

If S,A,jandCS,B,j

Then KD,Q^

R„:

If S,A„andCS,B,„

Then KD,C„

(49)

in which /?,y is the J-th rule in the i-th sub-rule base R., and /, denotes the number of rules in R-, therefore R, = U^v=0

[(Si-Aij ACSi.By) ^KDi.Cij]

(50a)

The overall rule base is comjKJsed of-/?,, i.e., R = {R„R„...R,,...,R„}

(50b)

Consequently, k'jj- =M,(s,,cs)xGK.=D{[(F(s)r\F(cs)]oR.}xGK.

(51)

Figure 2 schematizes the architecture of the FSMC. Note that for a n-statc system, a FSMC consists of n distinct two-input-one-output FLC as well as a sliding function evaluator (or called an input variables generator). Thus, the FSMC has 2n input variables.

290

Theorem

If the input space of each input variable of the FSMC is divided into m partitions, then the number of inference rules in a complete rule base for the FSMC is equal to / = nm^, where n stands for the number of system states.

proof: For a system containing n state variables, x„ x^, ..., x„, the rule base of the FSMC has n distinct sub-rule bases, namely, RB^, RB2,..., and RB„, and each sub-rule base RB- has two input variables, 5, and CS., as well as one output variable, AX),. Every sub-rule base RB. consists of/^ = m^ rules, / = 1, 2, ..., n, to form a complete subn

rule base. Therefore, n complete sub-rule bases have I = Y, li = nm inference rules. D For a conventional FLC, the complete fuzzy rule base for a system has n state variables, and that each of the state variables have m partitions, as inputs has m" inference rules. The above theorem indicates that by using the FSMC, the number of inference rules in a complete rule base is reduced to a linear function of system state n, instead of an exponential function of n for the conventional case. Therefore, the fuzzy-sliding mode control scheme performs with great effectiveness in the control of a large scale system.

2-tnpul'l-output fuzzy logic contiolle

RB: Rule Base. F:Fuzzifier.

DF: Defuzzifier.

PiPUnl

Figure 2. The FSMC architecture

291

MUX y^

Delay K"! _^___

^ RB: Rule Base F : Fuzzifier

^-R7>

P : Plant DF: Defuzzifier

MUX: Multiplexer DMU: Demultiplexer

Figure 3. The block diagram of FSMC with serial structure 3.4. Serial Structure of FSMC Generally, for a system with n state variables, one must establish n distinct sub-rule bases to form a whole rule base for the FSMC. In practice, the n distinct sub-rule bases can be established identically; therefore, one can establish only one rule base which has two linguistic input variables, S and CS, as well as one linguistic output variable, KD, with respect to the crisp input variables, (7sgn(x) and Aasgn(x), and the crisp output variable, k^^ , for all I = 1,..., «. The rule base might have the form /?;. IfS.Aj and CS.B^ Then KD. C.

j=h2,...,l

(52)

In this situation, the input variables are sequentially fed into the controller by a multiplexer, and the output variables are demultiplexed by a demultiplexer. Consequently, the structure of such an FSMC becomes a serial one as shown in Figure 3.

4 Simulation Examples and Applications 4.1 Simulation Examples In the following we illustrate three examples of second order systems and show how to develop the fuzzy-sliding mode control strategy. These examples are simulated by a digital computer with a sampling interval of 0.1 sec. and with the initial condition x(0) = [ 1 0 ] \ Example 1 Consider the system

292

x=

0 1 - 1 -2 _

X +

"0" 1

(53)

If we choose the sliding surface for the conventional sliding mode control as follows a = c'^x = [l l ] x and select ^o= 5. From Eqs. (33), (34), and (35), we have

k J = (c^b)'c^A = [-l-l] and

^; =

5, 0, -5,

for asgn(Xi) > 0 for asgn(Xi) = 0 for csgn(Xi) < 0

('•=1,2)

«^ = -ka^x = -£A:;j:,. Hence, the sliding mode control law can be expressed as follows: " = «., + « . = - ( C + k / ) X = (1-^;) X, + (1- k/) X, Figure 5 shows the response of the conventional sliding mode control systeiti. Next, let us design a FSMC for the system of Eq. (53). In designing the FSMC, we define the associated fuzzy sets to be shown in Table 1 and their corresponding membership to be shown in Figure 4, where we assume that (54)

Hs,.» =|is2.» = lis

|lc5,.*=Hci2.. =\i.cs

i*:P,Z,N)

(55)

and yiKD,.u=liKD,.u =\Ha)

i#: PB, PM, PS, Z, NS, NM. NB)

(56)

and their universes of discourse are all assigned to be [-6, 6]. The fuzzy-sliding mode control rules are described in Table 2.

293

5", .P: 5, is positive S,.Z: S- is zero Sj .N: Sj is negative CS^ .P : CSi is positive CSi .Z: CS^ is zero CS, .N: CS^ is negative

KD. .PB : KD. is positive big KD. .PM: KD. is positive medium KD, .PS: KD. is positive small KD..Z .KD.iszero KD. .NS: KD, is negative small KD, .NM: KD, is negative medium KD, .NB : KD, is negative big

Table 1. The fuzzy sets assignment of the FSMC (i = 1, 2)

IfS,.P2CRACS, .P Then KD,.PB If S,.Pand CS, Z Then KD,.PM IfS,.PsndCS, NThen KD:.PS If S,.Zand CS, PThen KD, .PS If S,.Zand CS, ZThen KD,.Z If S,.Zand CS, NThen KD,.NS If S,.Nand CS, P Then KD,.NS If S,.Nand CS, Z Then KD, .NM If S,.Nand CS, NThen KD .NB Table 2. The inference rules of the FSMC.

0.5 -

0.5

KO.PM

KD.PS

KO.Z

KD.NS

KD.NM

0.5

Figure 4. The membership functions of S, CS and KD

KD.NB

294

Set GS, =GS^ = 6 GCS^ =GCS^ =12 GK, =GK^ =0.85 The response trajectories of fuzzy-sliding mode control are shown in Figure 6. Comparing Figure 6 with Figure 5, one will notice that the chattering phenomenon in the fuzzy-sliding mode control system is smaller than that in conventional sliding mode control system. Example 2 To view the robustness of fuzzy-sliding mode control, let's assume that the system in Example 1 has paranrKter uncertainty as follows: X =

0 1 H - A ^ i -2-)-Aa2 J

\ +

0 I

where the parameter uncertainty, ha., i = I, 2, are uniformly distributed random processes with |Aa.| < 1. Applying the SMC and the FSMC given in Example 1, we get the response trajectories for the conventional sliding mode contr6l system and the fuzzy-sliding mode control system as shown in Figure 7 and Figure 8, respectively. Example 3 Consider the following nonlinear system: X =

0 1 -1 -2

X\ \X2\

Again, we apply the SMC and FSMC as given in Example 1, the response trajectories of both control schemes arc shown in Figure 9 and Figure 10, respectively. From the above examples, wc can make the following observations: 1. The dynamic behavior of a fuzzy-sliding mode control system is predictable by introducing a fuzzy-sliding mode regime. 2. The FSMC is robust against the plant parameter uncertainty. 3. In controlling the nonlinear system, the fuzzy-sliding mode control strategy also exhibits a good performance. 4. The chattering in the fuzzy-sliding mode control system is smaller than that in the conventional sliding mode control system

295

(c)

(d)

Figure 5. The response trajectories of conventional sliding mode control system (Example 1)

u

0.5 -

10

f.

5

o

-5 10

time, sec (c)

(d)

time, sec

Figure 6. The response trajectories oifuzzy-sliding mode control system (Example 1).

296

a

0.5

-10 time, sec

time, sec

Cd) Figure 7. The response trajectories of conventional sliding mode control system (Example 2) (c)

0)

0.5

time, sec

10

-5

10 time, sec (d)

Figure 8. The response trajectories of fuzzy-sliding mode control system (Example 2)

297


0.5

state, xl (.0)

time, sec

time, sec

(c)

(d)

Figure 9. The response trajectories of conventional sliding mode control system (Example 3)

•Z

0.5

-0.5

-5

(c)

time, sec

time, sec (d)

Figure 10. The response trajectories oifiizzy-sHding mode control system (Example 3)

298

4.2 Applications In this subsection, we will apply the scheme of FSMC to deal with two dynamic physical systems. One is a highly nonlinear inverted pendulum system, and the other is a two-link robot arm which has not only nonlinearity but also coupling effects. /. Inverted Pendulum System We will illustrate the performance of the FSMC by applying it to an inverted pendulum system as shown in Figure 11. In the inverted pendulum system, a cart which is controlled by an actuactor force/is free to move on a one-dimensional track. A rigid pole is mounted on this cart. The dynamics of the inverted pendulum system can be characterized by two state variables, with x, denoting the angle of the pole with respect to the vertical line, 6, and Xj denoting the angular velocity of the pole. By neglecting the coefficient of friction, the state Eq. can be expressed as follows [12]:

gsinxi-cosxi ""' " where

g m^ m / / -

f/

[-;;^xlsmxi-^;;^f^ 2f-cos2x,

the gravity constant, 9.8 m/s^. the mass of the cart, 1.0 kg. the mass of the pole, 0.1 kg. the length of the pole, 1 m. the input force to the cart.

In the design of FSMC, we treat the inverted pendulum as a black box with one input variable,/, and two output variables, Xj and JC^ , and we assume that x, and Jf^ are measurable. The control system is simulated by a digital computer with sampling interval At = 2 ms and initial condition x^{0) = 1 rad and XjCO) = 0 rad/sec. A. Regulation The regulation problem is to balance the pole in the vertical position by using the actuactor force/ to control the motion of the cart. So, the sliding surface can be chosen as CT = x, + X2 . By applying the FSMC as given in Example l,we get the response of the fuzzy-sliding mode control system as shown in Figure 12. To view the robustness of

/ T3

D"^

Figure 11. The inverted pendulum system

299

control system, one may assume that the pole has load variations and external disturbanc; the related results are shown in Figure 13 and Figure 14, respectively. B. Tracking Now, suppose that we want to control the states of the pole to track a desired sinusoidal trajectory given by x,j = a sin(M) x^j = dx,j / dt = aco cos(CiX) Then we can select the sliding surface as
+ (x2-xJ

Again, adopting the FSMC as given in Example 1, we get the system response as shown in Figure 15. Similarly, by adding the load variation and external disturbance shown in Figure 13b and Figure 14b, we get the resulting trajectories shown separately in Figure 16 and Figure 17. 2. Two-Join Manipulator Consider a two-joint manipulator shown in Figure 18, where g is the gravitational constant, 0, and 0j are joint angles, and u, and M^ are the applied torques. Let Xi = 6 i , X2 =6] , X3 =02, ^4 =02 and under the assumption that there exist lumped equivalent masses and massless links, then its dynamics can be represented as the following state space [20][22]: Xl

~ Xy

^ m v2.i.oft . . + ^ „y,g „ ^+. M,) . ^ -M f .t ^ x. ,2' .+ y,g + «,) Xl = -f(^,2>c^' + 2P,2A:2A:, ^ ( /- p X3 —X^

X4 = - --f-(Pi2^2' + 2p,jX^, + Y,g + M,) + -7-(-p,iX,' -I- Yzg + «,) with a,, = (m, + w^) / / -t- 7M^ /j^ + Im^ I, I2 cos Xj -H J, a,2 —m^ I2 + w^ /, I2 cos Xj Y, = -{{m, + m2)lj cos x^ + m212 cos(X; -I- Xj)) Y2 = - W j /j COS(X; + Xj)

300

0.5

5

10 tfme, sec

(b)

Figure 12 The responses of inverted pendulum system (Regulation) 0.4

0.2

5 (b)

10

time, sec

-50

-100 0.5 (c)

time, sec

1 Angle, rod

(d)

Figure 13 The responses of inverted pendulum system with load variation (Regulation)

301

Angle, rod

time, sec (c)

(d)

Figure 14 The responses of inverted pendulum system with disturbance (Regulation)

time, sec

Figure 15 The responses of inverted pendulum system (Tracking)

302

>
time, sec (c)

Figure 16 The responses of inverted pendulum system with load variation (Tracking) 2

> c

<

> c

< (c)

time, sec

Figure 17 The responses of inverted pendulum system with disturbance (Tracking)

303 Px2 = Wi 11 h sin X, where m,, I., J., i = 1,2 denote the point mass, link length, and additional constant inertia with respect to axis of rotation, respectively. This system not only has highly nonlinearity but also has coupling effects between two links. Let's assume that the parameters of the manipulator are given by: Wy = 0.5 kg, m^ = (l+w^) kg, /^ = 1.0 m, /^ = 0.8 m, and J,=J^ = 5.0 kg m^. Now, if we wish to control the manipulator to take a ball with the mass m^ = 3.0 kg from the coordinate (29/2, 0) to (-3"^, 0) and the initial angles arc 6,g= 1 rad, and O^,, = 0.5 rad, then, the desired paths can be exprssed as follows: path 1

path 2

palh 3

e„: e„ -^ jc/6 -> 57t/6 -^ e,g path 1

path 2

path 3

We can select three sliding surfaces for each path as follows: path 1: (Ty = \.5ix^ - 7t/6) +^2 CT^ = (JC3 + 7t/3) + X4

for link 1 for link 2

path 2:

for link 1 for link 2

a, = 0.8(Xi - 5KI6) + x^ aj = 0.5(x3-:i/3) + X4

path 3: a^ = 1.5(Xi - l)+x^ CTj = (X3 - 0.5) + x^

for link 1 for link 2

Owing to the insensitivity of the fuzzy-sliding mode, the coupling effects between two joints can be treated as external disturbance. So, we can apply the design procedure indicated in Example 1 to design two FSMCs for each joint independently, no matter whether these two controllers are identical or not. For simplicity, we use the same fuzzy set

Figure 18 The structure of two-joint manipulator

304

Figure 19 Response trajectories of 9, and 9^ assignments as shown in Table 1 with / = 1,2 ,..., 4, and their corresponding membership functions are the same as those shown in Figure 4. The inference rules in two controllers are the same as those listed in Table 2. Now, we select the scaling factors to be GS, = GS^, = 6, GCS^ = GCS^ = 12, GS, = GS, = 12, GCS, = GCS, = 24, GK, = GS^ = GK, = G A ; = 1 and (=1

i=3

with k,; = Mi a^gn(x)xGS,,

Aa^gn{x)xGCS,)xGK,

where i = 1, 2,..., 4, and k = 1 for i = 1, 2, and A: = 2 for i" = 3, 4. Figure 19 shows the response trajectories of 6^ and 6^.

5. Conclusions The proposed^zzv-i'/i(/mg mode regime gives rise to several characteristics as follows: 1. The dynamics behavior of the control system can be to a large extend dominated by a (fuzzy) sliding surface which is predetermined by the designer. In other words, like the classical sliding mode control system, the fuzzy-sliding mode control system is also insensitive to plant uncertainty as well as external disturbances. 2. Fuzzification of sliding surface will increase the robustness to the system nonidealities and decrease the chattering.

305

3. By observing switching control criterion of classical sliding mode control in a simple linear time invariant system, one can easily derive the fuzzy-sliding mode control rules. These fuzzy rules can be applied to the LTI systems as well as the nonlinear time variant systems. 4. Because the control feedback gain is obtained from the fuzzy inference rules, it is possible that the FSMC can control most of the complex ill-defined systems without knowing any of their mathematical model. Moreover, the sensitivity to plant uncertainty in the hitting phase of the classical sliding mode control will be attenuated in the fuzzy-sliding mode control. 5. The number of rules is a linear function of system variables rather than a exponential function in the conventional FLC.

Acknowledgements The authors would like to express their appreciation to Professor Hsin-Hsiung Lin of Tatung Institute of Technology for his careful reading and editing of the manuscript.

References [I] [2] [3] [4]

[5] [6] [7] [8] [9] [10] [II]

G. J. Klir and T. A. Folger, Fuzzy Sets, Uncertainty, and Information, Prentice-Hall: New Jersey, 1988. D. Dubois and H. Prade, Fuzzy sets and systems: theory and applications. Academic press: New York, 1980. L. A. Zadeh, "Fuzzy sets", Information and Control, Vol. 8, pp. 338-353, 1965. L. A. Zadeh, "Outline of a new approach to the analysis of complex systems and decision processes", IEEE Tran. on Syst. and Cyber., Vol. SMC-3, No.l, pp. 28-44, 1973. C. C. Lee, "Fuzzy logic in control systems: Fuzzy logic controller - Part I ", IEEE Tran. on Syst. Man and Cyber, Vol. 20, No. 2, pp. 404-418, 1990. C. C. Lee, "Fuzzy logic in control systems: Fuzzy logic controller - Part 11", IEEE Tran. on Syst. Man and Cyber., Vol. 20, No. 2, pp. 419-435, 1990. C. C. Lee, "A self-learning rule-based controller employing approximate reasoning and neural net concepts". Int. J. of Intel. Syst., Vol. 6, pp. 71-93, 1991. T. J. Procyk and E. H. Mamdani, "A Linguistic self-organizing process controller ", Automatica, Vol. 15, pp. 15-30, 1979. E. H. Mamdani, "Application of fuzzy algorithms for control of simple dynamic plant", PTOC. IEE. Vol. 121, No. 12, pp. 1585-1588, 1974. E. H. Mamdani, "Application of fuzzy logic to approximate reasoning using linguistic synthesis", IEEE Tran. on Computers, Vol. C-26, No.l2, pp. 182-1191, 1977. S. Assilian and E. H. Mamdani, "An experiment in linguistic synthesis with a fuzzy logic controller". Int. J. Man Mech. Studies, Vol. 7, No. 1, pp. 1-13, 1974.

306

[12] Y. Y Chen and T. C. Tsao, "A description of the dynamical behavior of fuzzy systems", IEEE Tran. on Syst. Man and Cyber., Vol. 19, No. 4, pp. 745-755, 1989. [13] R. Zhao and R. Govind, "Defuzzification of fuzzy intervals", Fuzzy Sets and Systems, Vol. 43, pp. 45-55, 1991. [14] M. Mizumoto and H. J. Zimmermann, "Comparison of fuzzy reasoning methods", Fuzzy Sets and Systems, Vol. 8, pp. 253-283, 1982. [15] V. I. Utkin, Sliding modes and their application in variable structure system, Moscow: Mir, 1978 (English translation). [16] V. I. Utkin, "Equations of sliding mode discontinuous system, I", Automat. Remote Control, No. 12, pp. 42-54, 1971. [17] V. I. Utkin, "Equations of sliding mode discontinuous system, 11", Automat. Remote Control, No. 2, pp. 51-61, 1972. [18] J. J. Slotine and S. S. Sastry, "Tracking control of non-linear systems using sliding surfaces, with application to robot manipulators". Int. J. Control, Vol. 38, No. 2, pp. 465-492, 1983. [19] J. J. Slotine, "Sliding controller design for non-linear systems". Int. J. Control, Vol. 40,No. 2, pp. 421-434, 1984. [20] K. K. D. Young, "Controller design for a manipulator using theory of variable structure systems", IEEE Tran. on Syst. and Cyber., Vol. SMC-8, No. 2, pp. 101-109, 1978. [21] K. K. D. Young, P. V. K. Kotovic and V. I. Utkin, "A singular perturbation analysis of high gain feedback system", IEEE. Tran. on Automatic Control, Vol. AC-22, pp. 931-938,1977. [22] K. K. D. Young and H. G. Kwatry, "Variable structure servomechanis design and applications to overspeed protection control", Automatica, Vol. 18, No. 4, pp. 385-400, 1982. [23] M. Vidyasagar, Nonlinear Systems Analysis, Prentice-Hall: New Jersey, 1993.

C H A P T E R 11

MULTIVARIABLE FUZZY SLIDING MODE CONTROL BY USING A SIMPLEX OF CONTROL VECTORS Giorgio Bartolini and Antonella Ferrrara University of Genova - Department of Communication, Computer and System Sciences Via Opera Pia, 1 lA-16145 Genova - Italy

1

Introduction

Since the basic work by Zadeh in the late sixties [12], [13], fuzzy logic has been intensively applied in the area of control system theory and design, in order to set up control environments which do not require the precise knowledge of the mathematical model of the process to be controlled (see, for instance, [3], [8] and the references therein cited). Indeed, fuzzy logic, enabling the definition of sets and membership functions of a certain clement to given sets by using natural language, appears to be suitable to deal with all those practical situations in which a certain level of uncertainty in the description of the system under study has to be taken into account. T h e design of a fuzzy controller consists of a fixed number of steps. T h e aim of the controller is that of relating the relevant variables, such as input or error signals, usually described in a crisp way, to the control action.

T h e first step is

therefore devoted to the conversion of the crisp input variables to the fuzzy form by associating with them the corresponding linguistic values. T h e core of the controller is represented by a set of linguistic rule. These latter are evaluated according to the compositional rule of inference, producing a fuzzy o u t p u t which, once transformed into a crisp variable, enables the generation of the suitable control action. In spite of the wide variety of applications of fuzzy controllers to industrial processes, a certain number of problems, mainly related to the efficient design of the look-up table containing the fuzzy rules, still remain to be overcome. A noteworthy a t t e m p t in this direction is constituted by the possibility of combining a controller design procedure based on fuzzy logic with a less qualitative design approach, such 307

S.G. Tzafeslas and A.N. Venetsanopoulos (eds.) Fuzzy Reasoning in Information, Decision and Control Systems, 307-328. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.

308 as that inspired to the variable structure systems theory, recently outlined in the literature [7]. A variable structure

controller (VSC) enables the definition of a suitable sliding

manifold on which the state trajectories of the controlled system are kept, from a certain time instant onwards, in spite of the uncertainties assumed on the process in question [10], [11]. T h e sliding mcinifold coincides with the intersection of the discontinuity surfaces associated with the various control signals acting on the plant. As a consequence, the overall control strategy, once the s t a t e trajectory of the controlled systems lies on the sliding manifold becomes discontinuous. T h e fuzzy version of the cleissical VSC proposed in [7] refers to single input/single o u t p u t plants and is characterized by a set of rules which enables the determination, at each sampling instant t = kAT,

k = 1 , 2 , . . . of the discontinuous gain vector to

be introduced in the state feedback loop to realize the conventional VSC scheme. Indeed, when the plant is single input, by using the VSC approach, it is often sufficient to guarantee that some simple upper and lowerbounds on system dynamics are known to attain the control objective with the desired accuracy. T h e control strategy is a signal, discontinuous on suitable manifolds of the state space, whose amplitude must be greater than some unknown but bounded function of the system states and disturbances. In the single input cetse, even when no upperbound of such an unknown function is known, a fuzzy controller can be designed since a simple qualitative relationship between the effect (the distance of state trajectory with respect to the sliding manifold) and the cause (e.g., an overestimation of the uncertainty bounds) can be stated, and, consequently, a suitable qualitative control action, based on rules of the type "the control action is too low, then increase it", can be adopted. It is worth noting, even in simple cases, t h a t this qualitative reasoning cannot be applied to any control technique. Consider, for example, the stabilization of a controlled system defined by

ij

=

xi + u

(1)

by using a piecewise constant control. We know that a control of such kind exists: for instance, if we consider a discrete time version of this system with zero holder sampling device, a suitable piece-wise constant control can be found. But we cannot easily represent such a control by using linguistic rules of the type "if i j is high positive then u be high negative".

309

Therefore, when dealing with the possible use of a fuzzy system to control complex uncertain plants, it is important to choose a conventional (crisp) control strategy which is suitable for being described by means of finite number of fuzzy rules (the Ziegler-Nichols method for tuning the parameter of a P.I.D. regulator belongs, for example, to this class of control techniques). In this chapter, we move from the single input case and focus our attention on the multivariable control of non linear uncertain systems. In particular, a fuzzy controller ba^ed on a particular version of multi-input VSC strategy will be dealt with. To this end, some assumptions on the class of controlled systems needs to be made. More precisely, we shall confine our investigation to the class of systems modelled by pth-order controlled differential equations on 3?"". We assume that the full state is available for feedback purposes and introduce weak a-priori assumptions in order to guarantee the existence of the solution to the control problem. In order to motivate our choice, let us consider a controlled m-degrees of freedom mechanical system

A{q)q = u + fit,q,q)

(2)

where q{i) £ 3?"' is the vector of the generalized coordinates, A{q) is the inertia matrix, f{t,q,q)

represents elastic, damping, friction, Coriolis and other torques, us

well as external disturbances. If additional constraints are not included and no coordinate transformation has been performed, it is well known that A{q) is symmetric and positive definite. The robust control of this kind of system, as far as both regulation and tracking problems are concerned, hcis been addressed by Utkin [10], [11] and, more recently, by Guzzella and Geering [5] and Slotine and Li [9]. The proposed procedure is worth being briefly recalled in order to put in evidence how the relevant control structure is suitable for being inserted in a linguistic (fuzzy) control environment. To this end, consider the m-vector s

-

q + Cq

s'^ - [si ... Sm]

C

=

diag{Ci}

C, > 0 i = l , . . , m

On the manifold s = Q the behaviour of the system is equivalent to that of m decoupled asymptotically stable first order linear systems. Therefore, we can reformulate the regulation problem as follows: "make the trajectories of the controlled system reach in finite time the sliding manifold and there remain for the remainder of the control interval".

310

In order to attain such an objective, consider the scalar function V = \s''A{q)s

(3)

Since A{q) is positive definite, V satisfies the basic assumption to be regarded as a Lyapunov function candidate in 3?"". Its time dervative is V = s'^A{q)s + s'^A(q)s

(4)

From the theorem of energy conservation A{q) = K{q,q)^K''{qA)

(5)

where A' is a matrix such that K{q, q)q represents the m-vector of centripetal and Coriolis torques. As a consequence, V- = 3^ {« + / ( ^ 9, q) + A{q)Cq + [A'(9, q) + K^{q, q)] {q + Cq)]

(6)

that can be rewritten a^ V = s'[u + M{t,q,q)] If a known upperbound M'{t,q,q) then a control law of the type

of the norm of the vector M{t,q,q)

Ui =-M'(t,q,q)sign{si,}

i = l,...,m

(7) is available,

(8)

causes the reaching, in finite time, of the sliding surface s = 0 and the fulfilment of the control objective. This result is well suited for being dealt with in a fuzzy control environment. Indeed, suppose that M* is not known in advance. It is clear that any control component can be represented as Ui =

-jisign{si}

Then, a set of linguistic rules can be individuated relating the effect, the component Si, and the cause, the choice of the parameter 7,, in order to generate a control causing the system trajectory to reach the sliding manifold (e.g., "if Si is positive big, then the increment of 7, is positive big"). Note that this fuzzyfication procedure might fail if different approaches, like feedback linearization [6], were used, since.

311

in such cases, the control law strictly depends on the equations which describe the controlled system dynamics. On the other hand, this approach cannot be pursued if the generalized coordinates are subject to a non linear trasformation X =

F{q)

X =

J{q)q

(9)

where J{q) is the Jacobian matrix. In that case, the differential equation governing the evolution of the system in the new coordinates, assuming the transformation to be invertible, becomes X =

J{q)q + J{q)q

X =

J{q)[A{q)]-'[u^

f{t,qA)\-^

J{q)q

(10)

which can be rewritten as A{q)[J(q)\-'x^Y{t,q,q)^u This equation is similar to (2), but this time the matrix A{q)\J(q)\~^

(11) does not

result symmetric and positive definite (j4(9)[J(g)]~' is however assumed to be invertible). Therefore, the previous control procedure cannot be applied. In the next sections it will be shown how it is possible to use a suitable control procedure, the so-called simplex sliding mode control method, in order to deal with systems of the type (11) in a fuzzy context. This chapter is organized as follows. Section 2 contains a presentation of the simplex method for the control of multi-input uncertain systems. Section 3 is devoted to the description of the fuzzy control evironment in which the problem considered is inserted, as well as to the presentation of an illustrative example to complement the theoretical discussion.

2

The simplex method for the control of multiinput non linear systems

T h e objective of this section is that of illustrating the main results of the method proposed by Baida and Izosimov for the control of multi-input non linear systems [1], and to generalize them in order to cope with uncertainties on system dynamics.

312

Starting from their results, a new control algorithm is proposed which proves to be suitable to constitute the control philosophy underlying the fuzzy controller which will be presented in the next section. Consider a system described by the following differential equation x = A{x) + B{x)u

(12)

where x € 3i", u € 3?"*. We choose a manifold {s(x) € 3?"" : ^(x) -Cx -0} so that when the system trajectories lie on it (i.e., i(x) = 0, if s{x) = 0), the behaviour of the system satisfies a prespecified control obiective. The control signal role is, therefore, that of forcing the system trajectories from any initial state to reach the manifold in finite time. Mathematical and applicative details of such an approach can be found for instance in [2], [4], [10], [11], [14]. In order to provide a control strategy accomplishing the above control objective, let us consider the dynamic behaviour of the m-veclor s{x) described by s(x) = C A{x) + C B(x)u

(13)

where the matrix [CB{x)] is assumed to be invertible, but neither symmetric nor positive definite. The traditional way of extending to the multi-input case the basic VSC strategy consists in the selection of the control signals according to a componentwise procedure. This means that each component of the vector s{x) is related to the corresponding time derivative i(x) by the so-called reaching condition, i.e., Si{x) = -7?sijfn{si(x)}

(14)

s{x) = —rsign{s{x)}

(15)

that is where F := diag{%}, i = 1 , . . . , m The control vector results to be, u = [CB(x)]-' [-Tsign{s{x)}

- CA{x)]

(16)

Note that each component of vector u is a linear combination of discontinuous signals. This fact prevents the plain application of the componentwise procedure when some uncertainties affect B{x) and A{x). Moreover, in that case, it is not clear if there exists a simple way of changing the components of u according to qualitative rules in order to satisfy, even in the uncertain case, the reaching condition.

313

However, the reaching condition can be expressed also in alternative ways, giving rise to different control switching logics. Among these, the simplex method will be exploited in this chapter in order to transfer the conventional sliding mode control method into a fuzzy context. This procedure can be briefly described as follows. 1. Choose in 3?*" a simplex of vectors [ui, . . . ,Um+i] containing the origin, i.e.

2. Partition S™ in m + 1 non overlapping regions m+l

3. Perform the following transformation y

=

y

=

lCB{x)]-h(x) [CB{x)]~'s{x)

+ j^{[CB{x)]-'}s{x)

(17)

that is y

= +

{CB{x)]-'CA{x) E

^^^fj.^^''

IM^)

+ B,{x)u]six)

+ u

(18)

where -4,(x), Bi{x) are the rows of A{x), B{x), respectively. Then, y - g{x,s,u) Note t h a t we assume that —g(x,s,u)

+ u

(19)

= 5I)]^"i^' ^iWi, I ^ i l t ' ^i = Qo < 1, which

means that the equivalent control «o = —g{x,s,u)

belongs to the convex set gen-

erated by the simplex vectors. Such an assumption holds if the amplitude of u, is sufficiently high and the values of .5(1) is sufficiently close to t h e origin (see (18)). In the paper of Baida and Izosimov two theorems have beeen proved concerning the choice of the control switching logic which guarantees the convergence to the origin of the vector y in finite time [1]. Since [CB{x)] is assumed invertible the convergence of s(x) to the origin is also guaranteed. These theorems will be presented without proof.

314

T h e o r e m 1. Partition

the y-state

space in m + 1 non overlapping

n , = J J/ 6 »"• : y =

"^

regions

XiUi, A. > 0 I

If the control is chosen as u = Uj, when y G ilj, then the origin y = 0 is cally stable. Moreover, oj the convergence

the convergence takes place in finite time and an

asymptotiupperhound

interval t' — to is given by I

-ao

R e m a r k : the control is unique for any region, and any region is univocally determined by the choice of the simplex of control vectors. T h e o r e m 2. Partition

the y-state

state space in m + I non overlapping

( If the control is chosen asymptotically upperbound

stable.

m+l

regions

]

J/ e »"• : t/ = - J2 - ^ i " - ^ ' ^ ^ \ to be u = u,-, when j / € 17^, then the origin y = 0 is

Moreover,

of the convergence

the convergence

takes place in finite time and an

interval t* — to is given by 1 - ao E7=, A,(y(
R e m a r k : This control strategy is not unique because for each fJj we can choose one control vector among the set {«,}, i ^ j . To solve such an ambiguity in [1] it is proposed to choose a control sequence of the type

f

Ui+i i = 1, . . . , m ui

1= m + l

Once this choice has been made, a unique correspondence between the choice of the simplex of control vectors and the partition of the y - s t a t e space is stated, and a unique relationship between any region of the y state space and one of the control vectors is established. [CBix)].

All this facts require the perfect knowledge of the matrix

315

Q

Q.

Figure 1: T h e regular simplex of control vectors Indeed, if [CB(a;)] is unknown, though invertible, while s is available y = [CB{x)]~^s

in unknown and it is impossible either to know which region y belongs

t o , or to define the correspondence between the unknown regions and the control vectors. This fact prevents the use of the convergence theorems previously indicated in their present form. It is however possible to show t h a t , even in the uncertain case, t h e convergence to the sliding manifold of t h e s vector can be obtained by means of elementary operations on a fixed simplex of vectors. For elementary operations we mean rigid rotations of the regular simplex and increments in the modula of the component vectors. T h e rationale of our procedure will be graphically illustrated by an example in 3?^.

Let us consider a partition in the 5-plane induced by a regular simplex of

control vectors. In Fig. 1 the regions il,^ = {s £ di^ : s = Yll=i,i^j ^i^h are explicitely indicated. By means of the invertible transformation y =

^i > 0} [CB{x)]~^s

any simplex in the s-space is trasformed into another simplex in the y-space. If det[CD{x)]

is positive, this latter is, in general, irregular and results to be rotated

with respect to the starting one. If det[CB(x)]

is negative, but it does not change

sign for any a: 6 A' C 3?", the simplex of the transformed vectors can also be permutated. det[CB{x)]

T h e permutation is however invariant within the domain in which

does not change sign. Therefore, such a p e r m u t a t i o n can be determined

by evaluating it in an isolated point x £ X. hereafter, only the case det[CB(x)]

T h e n , for the sake of simplicity,

> 0 will be considered. A possible situation is

shown in Fig. 2. It can be noted t h a t , in general, starting from a regular simplex of control in the s-plane it is not ensured that this simplex constitutes a control

316

n Figure 2: The irregular simplex strategy in the j/-plane according to the previously mentioned theorems. Indeed, from the uniqueness of the control strategy estabilished by such theorems the control triple {u),U2,U3}, should coincides with the triple {w'j,1X2,^3} which is obtained as u'- = [CB{x)]~^Ui

and consequently turns out to be an irregular simplex. Note that

in Fig. 2 the regions HJ, i = 1,2,3 are referred to the transformed control vectors. However, by means of a rotation of the regular simplex it is possible to locate the narrower cone of the irregular simplex inside the region defined by the corresponding vectors of the regular simplex (see, for example, Fig. 3 where u[ = Yll=i '?i,"ii " i = E?=i'/2."n Vu, V2, > 0). In this case, if the trajectories are directed as the corresponding vectors Ui, Wj, U3 (i.e., the uncertain vector g is assumed to be zero), the strategy w = Uj if 3/ G fi,, u = U2 if 2/ € ^21 w = W3 if J/ € Hg causes y to slide towards the origin. Our aim is that of proving that for any partition of the j/-space into TTI + 1 regions by means of an irregular simplex, it is still possible to a.ssociate with each of such regions a suitably oriented control vector belonging to a regular simplex which causes the origin of the j/-space to be reached in finite time. The accomplishment of this tcisk can be divided into two steps corresponding to the following two theorems. In the next section, this result will be used as a basis for the development of a fuzzy control approach in which a regular simplex is rotated and modified in amplitude by using, instead of the unavailable vector 3/, the measurable vector s, until the desired convergence to zero takes place.

317

Figure 3: Correct position of t h e narrower cone of the irregular simplex T h e o r e m 3 . Partition

the y-state

space in m + 1 non overlapping m+i

regions

1

! under the assumption

previously

indicated for g^x^s^u).

If the control is chosen

as u = — Z!{^t'>!i V'"i) Vi ^ 0) JTi^t'tViVi ^ If w/ien j / G ilj, asymptotically

stable and the convergence

P r o o f . Consider y = J2^\-tj

takes place in finite

then the origin is

time.

A,Ui, and, for the sake of simplicity, assume t h a t A,-

are all greater than zero. Then, define A :=

[Ai . . . Aj_] A j + i . . . An,+iJ

A =

[[/,]-• 3/

where [(/,] := [ui ...iij_iUj+i ...Um+i]

(20)

is an invcrtible m x m matrix. The control

signal can be expressed as m+l

u

=

-

Yl

u

=

-[U,]4>

V."M

0 (21)

where ^ ^ := [v?, . . . i^j-iV'j+i . . . v „ + , ] . From the above definitions, one obtains A = [U,]-' y

{22)

318

with y — g ( i , s , u ) + u, as indicated in (19), where, by assumption, —g{x,s,u)

=

Z)I^t' ^,w,, cind X^-^t' h = OiQ <\. Thus, we can rewrite - 9{x, s, u) = [Uj] [fci . . . kj^ikj^i

... k,r,+iY + ^j"j

(23)

where Uj = — J^J^t.^i/'•"•' f^> — ^' ITI^t.^^j/*• > ^> since the control vectors u^'s form a regular simplex. Therefore, defining Kf

:=

[ki... kj.ikj+i...

Mf

:= [fti...fij-ifij+\

i„+i] • • • /«m+i]

it yields - g{x, s, u) = [U,] K, + k, [[/,] M,

(24)

so that, from (22), A =

m-'{-[UAK,-kAU,]M,-[U,]
=

-Kj - kjMj -

(25)

Now, consider the scalar m+l

a(2/,0=

E

^"

A. = -^.(2/.'),

Q(0,0 = 0

(26)

1=1,iVj

with ot(y,t) > 0 for 2/ ^ 0, and Q(y,f) —> oo for | y — | + oo. It satisfies the conditions to be a Lyapunov candidate for system (22). Summing row-by-row equations (25), one has m+l

m+l

m+l

E h-ki X: /^.- E ^i

a=-

1 = 1,i/;

i=l,i?'j

(27)

• = lii?'j

Then, m+l

/

m+l

\

m+l

a = - E ^' + ^i h - E /'i - E V. i=l

<

\

i=l,i5^J

/

•=!.;#>

-(1+ao)

(28)

this implies that a(-, <) = 0, for any t > t', with t' = o(j/o, 0)/(l + QQ) D

Then, we are in a position to state our main result, which generalizes the previous considerations about the possibility of using a regular simplex of control vectors

319

when the state space is partitioned on the basis of an irregular simplex of vectors. T h e o r e m 4 . Ifm

vectors [u\ . . . u'j-i^'j+i • • • "m+i] of an irregular simplex belong to

a region ilj of a regular simplex,

that is

m+l

"i =

X) '?•"•' '/'^ <^' fc = l , . . . , j - l , j + l , . . . , m + 1 i=l,.-5ti

then, the control

strategy u = Uj if J/ e Cl'j

where

causes the finite time reaching of the origin of the y-state

space.

P r o o f . If y belongs to Cl'j, which is contained in Clj, by applying Theorem 1, it is easy to see t h a t the control action u = Uj forces y to reach either the origin or the boundary of f l ' . In the other regions, any vector of the regular simplex satisfies the condition of Theorem 3 with respect to the irregular simplex, that is any Uj can be expressed as Uj = — Z^i^t'i/j A^ii with Yl^]i:^j

0i > 1- Hence, in any case, y

reaches the origin in finite time. D

3

T h e proposed fuzzy controller

T h e aim of the present section is that of transferring the m u l t i - i n p u t simplex sliding mode control approach and the results proved in the previous section into a fuzzy framework. Indeed, this way of proceeding seems particularly suitable to deal with all those situations in which a certain amount of uncertainty affects the mathematical model of the system to be controlled. In the previous section, it was pointed out how a regular simplex of control vectors can be modified (that is, it can become irregular and be rotated with respect to the original one), because of the application of a transformation matrix to the various component vectors. In Theorem 4 it was however indicated a possible way of positioning the modified simplex, with respect to the regular simplex, in order to guarantee the finite time reaching of the origin of the y - s t a t e space. T h e design of

320

a fuzzy controller solving the control problem in question can be based on such a theorem. When the transformation matrix is unknown, it is not possible to quantify the actions to be applied to the regular simplex in order to properly re-positionate the irregular simplex with respect to the previous one. Such actions consist in rigid rotations of the simplex and variations of the modula of the component vectors. T h e n , it seems natural to establish a qualitative relationship between the action to perform and the difference between the actual and the expected behaviour of the system. Suppose to choose, as a switching logic, that expressed by Theorem 1, with the difference that now vector y is substituted by vector s. Because of the transformation operated by matrix [CB(x)],

it is not ensured that 5 moves towards the origin of

the s-space. On the other hand, if the disturbance g{x,s,u)

in (19) is small and

a regular simplex of vectors with suitable amplitudes is applied, the trajectory of the unknown vector y is approximately a straight line parallel to the corresponding applied control vector, according to (19). While y is unknown, the vector s(x) [CB(x)]y

is measurable.

=

This latter moves along a trajectory which, as long as

[C/?(j;)] were constant, would result in a straight line as well, if evaluated in a short time interval, the angle between this line and the direction of the applied control vector being the opposite of the angle of which the regular simplex needs to be rotated in order to have a motion of s directed in the desired way. In this situation, defining As{t)

:= s{t) — a{t — At), the action to be applied to the regular simplex is

just a rotation of an angle 0(t) = arcos[< Aa{t),u

> /(||As(i)||||u||)].

Since g{x, s, u) is not known and [CB{x)] is not a constant matrix, the above procedure cannot be applied in a quantitative way, unless some a-priori known bounds on g{x,s,u)

and [CZ?(x)] are available. Also in t h a t case, as previously outlined,

some problems remain to be faced. Indeed, when the uncertainties are described in terms of interval vectors and matrices, the possibility of losing the invertibility of matrix [CZ?(a;)], as well as the necessity of performing a highly conservative control design, should be taken into account. On the contrary, by using a qualitative approach, based on the experience that the amplitude of the control vector is required to be sufficiently high to dominate the unknown disturbance g(x,s,u), to the angle (f> between As{t) to overcome these drawbacks.

and that the rotation angle 0{t) has to be tied and u for sufficiently small At,

it seems possible

It should be noted t h a t , according to the results

expressed by Theorems 3, 4, the ideal value of the angle 0{t) is not unique, but can

321

be expressed as an interval of values such that for any 0(t) belonging to that interval t h e finite time convergence of s to the origin takes place. This implies that when a contraction of the norm of s occurs (i.e., A||s(<)|| = ||'S(OII - ll^(' — ^011 '^ negative) the positioning of the simplex of control vector could be stopped. In order to design the fuzzy controller according to the previous considerations, we proceed in the standard way, selecting the suitable variables to be used a.s terms in the premises of the rules. Then, the design procedure can be performed according to the following steps: 1. Fuzzify the relevant variables (input variables to be included in the premises of the rules, output variables on which the consequence of each rule is built), defining the fuzzy sets, in their universe of discourse, associated with them, and the corresponding membership functions. 2. Construct the decision table by selecting a suitable set of rules. 3. Determine the membership function corresponding to each rule and the overall membership functions associated with the output variables. 4. Deffuzzify the output variables obtaining the crisp values of the control actions. The previous list can be particularized to the control case considered within this chapter, in order to obtain a fuzzy simplex sliding mode control scheme for m u l t i input systems. T h e input variables to be considered can be the amplitude of vector s, the variation of the amplitude (i.e., vector A||s||), and finally the rotation matrix between vector A | | s | | and the currently applied control vector (w = Uj if s £ ilj).

While ||s||

is available at any time instant, the other quantities are supposed to be evaluated at discrete t i m e intervals kAt,

k = 1,2,

Note t h a t the sampling interval At could

be regarded as an input variable, as well. T h e o u t p u t variables are the rotation matrix Rk which needs to be applied to the previous control signal, which we call Wk-i, in order to determine the current control Wk according to the following relationship Wk

=

RkWk-l

=

RkDkU

Do

=

I,

Ro = I

Dk

=

Rk-iDk.i,

k = l,2,...

(29)

322

and the amplitude Au^ of the components of the simplex of control vectors. Both the input and output variables are fuzzified by associting with t h e m suitable fuzzy sets corresponding to a prespecified number of levels, with proper membership functions. T h e above considerations can be clarified by means of a simple example relevant to a two-input system described by the following state equation

s = Hu+ g

(30)

ith

H-' =

- 0 . 5 ( 1 +7sen*i/(a))

-0.866

-0.866(1+7senV(.s))

-0.5

a {I +/3sen^i/(5)) 9 =

a {I +

0sen^i/{s))

with iy{s) := (1 + s^ + ^2)'^^- Note t h a t if 7 :^ 0, matrix / / " ' is non symmetric. T h e control vectors constituting the initial regular simplex are uj = [0. 10.] ul = [8.66 - 5.] ul = [-8.66 - 5.] T h e unknown vector y = H~^s evolves according to the following differential equation y = H-h

+ H-^g-\-u

(31)

T h e proposed fuzzy controller is based on the rules shown in Table 1 which refers to the membership functions relevant to the input variables ||s||, Ajl^jl, 0, and the o u t p u t variables 9 and Au indicated in Fig. 4, 5, 6 and 7 and 8, respectively. T h e behaviour of the controlled variable 3, starting from the initial condition s^ = [2, 3], is illustrated in Fig. 9 under the assumption that 7 = 0, o = 0 and constant modulus of the applied control vector (i.e., A M = 0, which means that the last column of the rule table is not considered). In Fig. 10 the behaviour of vector 6 is reported assuming, instead, that 7 = a = 1, ^ = 0.5, and allowing the adjustment of the modulus of the control vector u. In both cases the behaviour is highly satisfactorily, since the convergence of the controlled variable to the origin of the state space is obtained with good convergence rate in spite of the uncertainties on the non linear model of the controlled system.

4

Conclusions

In this chapter a fuzzy control approach is presented in order to deal with multiinput uncertain non linear systems.

323

Figure 4: Membership function for \\s\

I

Figure 5: Membership function for A||5|

324

phi

Figure 6: Membership function for 4>

!

Figure 7: Membership function for 9

325

20

25

30

control magnitude

Figure 8: Membership function for Au

I a

-1

0

1

Componeni 2

Figure 9: The controlled variable s, 7 = 0

50

326 I

!

!

I

—1

-

5

-

4

-

3

-

2

-

1

0

1

I

2

3

4

5

Coinponenl2

Figure 10: T h e controlled variable 3, 7 = 1 The starting point is the choice of a control algorithm which does not result to be too strictly dependent on the perfect knowledge of the mode! of the system to be controlled. T h e most effective approach in this sense, among those proposed in the literature, is the so-called simplex sliding mode control method, in which, however, the transformation matrix acting on the control vectors is required to be perfectly known. In the present chapter it has been shown that this requirement can be relaxed as long as a non conventional control approach is applied. T h e resulting algorithm turns out to be characterized by the application of simple operations on vectors, like rotations and increment of their modula, which can be easily translated in terms of fuzzy rules, giving rise to a new fuzzy controller which enables to solve in a satisfactory way the crucial problem of controlling uncertain non linear systems.

327

If ||.|| is ... BIG BIG BIG BIG BIG

POS POS POS POS POS

BIG BIG BIG BIG BIG

POS POS POS POS POS

BIG POS BIG POS MED POS MED POS MED POS MED POS MED POS MED POS MED POS MED POS MED POS SMALL POS SMALL POS SMALL POS

and All^ll is ... and 4> is ... then 9 is ... POS POS POS POS EZ EZ EZ EZ

HIGH MED LOW EZ HIGH MED LOW

EZ HIGH MED

LOW MED LOW LOW EZ HIGH MED

LOW EZ HIGH

LOW LOW MED

MED LOW EZ

LOW LOW LOW EZ MED LOW EZ

EZ

NEG NEG NEG NEG POS POS POS POS EZ EZ EZ EZ NEG POS EZ

HIGH HIGH MED LOW HIGH MED LOW

HIGH MED LOW

NEG Table 1: The rule table

and Au is ... BIG POS SMALL POS BIG POS BIG POS

328 References [l] Baida, S. V., and D. B. Izosimov 1985, Vector method of design of sliding motion and simplex algorithms, Automation

and Remote

Control, 4 6 , 830-837.

[2] Bartolini, G., and T. Zolezzi 1985, Variable structure non linear in the control law, IEEE

Transaction

on Automatic

Control, 3 0 , 681-684.

[3] Chang, S., and L. A. Zadeh 1972, On fuzzy mapping and control, IEEE

Trans-

action on Sys., Man, Cyber., 2, 30-34. [4] DeCarlo, R. A., Zak, S. H., and G. P. Mattews 1988, Variable structure control of non linear multivariable systems: a tutorial, Proceedings

of the IEEE,

76,

212-232. [5] Guzzella, L., and H. P. Geering 1990, Variable structure controllers for robots, in Deterministic

Control of Uncertain Systems,

A. S. I. Zinober Editor, Peter

Peregrinus, London. [6] Isidori, A. 1985, Non linear control systems, Springer-Verlag, Berlin. [7] Kung, C. C., and S. C. Lin 1992, A fuzzy sliding mode controller design, Proc. IEEE Int. Conf. on Systems

Engineering,

Kobe, J a p a n , 608-611.

[8] Lee, C. C. 1990, Fuzzy logic in control systems: fuzzy logic controller, Transaction

IEEE

on Sys., Man, Cyber., 2 0 , 404-435.

[9] Slotine, J. J. E., and W. Li 1991, Applied non linear control, Prentice-Hall, Englewood Cliff, N.J.. [10] Utkin, V. I. 1978, Sliding modes and their application in variable structure systems, MIR Publishers, Moscow. [11] Utkin, V. I. 1992, Sliding modes in control and optimization, Springer-Verlag, Berlin. [12] Zadeh, L. A. 1965, Fuzzy sets, Informat.

Control, 8, 338-353.

[13] Zadeh, L. A. 1968, Fuzzy algorithms, Informat.

Control, 12, 94-102.

[14] Zinober, A. S. I. (editor) 1990, Deterministic Control of Uncertain Systems, Peter Peregrinus, London.

CHAPTER 12 KNOWLEDGE REPRESENTATION AND INFORMATION P R O C E S S I N G IN I N T E L L I G E N T C O N T R O L L E R S

Kaoru Hirota<*) and Witold Pedrycz(**) (*) (**)

Dept. of Instrument & Control Engineering College of Engineering, Hosei University Koganei City, Tokyo, 184 Japan Dept. of Electrical & Computer Engineering University of Manitoba, Winnipeg, Manitoba, Canada, R3T 2N2

ABSTRACT Pure numerical computations have been playing a dominant role in control systems. With the increasing complexity of the systems under control and diversity of control tasks to be handled, it became obvious that the concept of this numerical platform mastered so far cannot be sufficient in many situations. Some techniques of Artificial Intelligence (such as e.g., control expert systems) on one hand, and nonparametric parallel optimization (i.e. structures such as neural networks) on the other, have been already studied as possible computational frameworks. Following this avenue, we will investigate the problem of knowledge representation and information processing through the use of an integrated platform composed of neural networks and fuzzy sets. This unified approach takes advantage of flexibility of fuzzy sets used in representing control knowledge and learning capabilities residing within neural networks. The proposed topology of the controller includes an internal logic-based part describing control rules and an interface component "translating" these rules and their activation levels into pointwise control actions using either preselected (fixed) or trainable control values. With regard to this feature the learning of the controller is reinforced or is carried out in a fully supervised mode. KEYUORDS Knowledge representation, fuzzy sets, reinforcement learning, neurocomputations, fuzzy controller 329 S.G. Tzafestas and A.N. Venetsanopoulos (eds.) Fuzzy Reasoning in Information, Decision and Control Systems, 329-343. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.

330

1.

INTRODUCTION

Witnessing a rapidly growing field of control of complex systems, it becomes evident nowadays that knowledge representation issues are essential to the development of new control strategies. In other words, representation and processing aspects of control should be considered as equally significant in designing intelligent controllers. Usually much more attention has been devoted to purely numerical knowledge processing while relatively less research effort has been paid to address properly all relevant facets of representation of the underlying control knowledge. The plain numeric processing has occupied a central role in the control area for many decades. Along with the increasing complexity of systems and a lack of knowledge pertaining to their performance, some alternative avenues have been pursued. Fuzzy sets [12] [14] [15] [16] and paradigms of expert control [1] on one hand, along with emerging computational power of neurocomputations on the other, can form an interesting and powerful designing platform. To highlight their most useful properties we will commence with a detailed discussion on the crucial features of intelligent control. The objective of this presentation is to clarify the role of fuzzy sets and neural nets in intelligent control as well as to highlight their mutual relationships along with a complementary character in their performance. It has already been recognized that all those elements may work efficiently, refer e.g., to a special issue of the IEEE Control Systems Magazine [4] devoted to neural networks used for system modelling and contfol. Subsequently, we will come up with a more detailed discussion concerning topologies of the controllers pointing out how the introduced concepts are blended into a coherent architecture. A particular attention will be focussed on a suitable partition of tasks of knowledge representation and numerical processing as they are realized by fuzzy sets and neural networks.

2.

PRiriARY FEATURES OF INTELLIGENT CONTROL

In intelligent control we can envision two prevailing attributes leading to a clear distinction of this type of control from a vast number of available control algorithms. The first one concerns the development of the suitable, well-handled, and manageable perspective on the control strategy. In its principle, the perspective should assure that all the system and control variables should be perceived at a suitable level of precision that is necessary to capture the most intrinsic features of the domain knowledge of the control problem at

331

hand and visualize essential relationships between the variables. Rather than concentrating on mastering linear or nonlinear local relationships between all the variables again handled in a numerical pointwise manner, it might be useful to move toward a higher level of generality. This level can tackle with information granules defined for involved variables. By this type of information aggregation we mean a collection of associated values which are coupled to a certain degree within the given context of the specified control task (control situation) and labelled for further use. From a conceptual standpoint we can indicate three distinct approaches already reported in hterature. (i) the first one, usually residing within the area of Artificial Intelligence (AI), refers to purely symbolic computations. The aggregates studied there and being almost of a Boolean character, are named symbolically and used afterwards. Possessing a certain syntax, these aggregates being processed as purely symbolic quantities give rise to a so-called Qualitative Modelling (Q^) and Qualitative Control, (QC), cf. [2] [3]. Obviously, the very fundamental nature of the AI techniques does not leave any room for numerical processing. In many control situations this approach cannot be fully accepted mainly due to significant and irreversible loses of information caused by this symbolic quantification, (ii) the second approach deals with aggnsgates having a form of closed numerical intervals (as well as hypercubes forming their multidimensional counterparts). These quantitites create a basis for the development of a certain cognitive perspective at which the control processes are looked at. In this setting, the existing techniques of interval calculus widely apply, cf. [5]. This way has been exploited in the past mainly for purposes of performing much more thorough sensitivity analysis and modelling systems using sel-theoretic perspective (e.g., set-theoretic models with unknownbut-bounded disturbances). Within the above framework numerical processing capabilities are pnsserved. The increased generality is obtained by adjusting (increasing) the size of the information aggregates (granules) through a suitable increase of the widths of the intervals. Definitely, the control algorithm becomes robust due to the introduced interval-valued character of the variables. The obvious shortcomings manifesting here is that border values of the intervals might be difficult to establish. The interval-based knowledge representation one can easily control the level of specificity (generality) of the concepts. It is, however, not capable of distinguishing between the values (points) belonging to the same interval. In the third approach we will consider a combination of (I) and (ii) that is carried out as follows.

332

(iii) Referring to the lack of the focussed concentration (selective discrimination) occurring in the previous framework we will replace intervals by fuzzy sets. Fuzzy sets can eliminate the above mentioned disadvantage due to their membership functions acting as elastic constraints and highUghting a degree of membership to the given granule. They can carry out a detailed discrimination process within the same scope of cognition therefore properly addressing the question of the selective focus of attention. The higher the grade of membership, the higher satisfaction of the given property (constraint). Subsequently, some elements in the granules become strongly elevated while the others substantially suppressed. An immediate incorporation of fuzzy sets in QSi and QC as introduced e.g. in [2] allows to cope with the semantics of the conveyed terms and take care of well-defined syntax features. Fuzzy sets, as generally set-theoretic oriented concepts, are not extremely powerful at numerical computations. The collection of the existing and commonly utilized operations is not well suited for effective and fast calculations. These operations are not capable of coping efficiently with a probabilistic orrepetitivecharacter of information. In designing fuzzy controllers, the general control strategy is synthesized on a basis of some individual and scattered pieces of knowledge such as e.g., the previous records of control acdons or more or less complete control protocols. Sometimes these protocols are biased by a lack of precise data which are replaced by some linguistic statements. These situations call for the use of fuzzy sets as a relevant formalism to cope with linguistic and/or incomplete information. There are many diverse conceptual as well as applied platforms which can directly benefit from the methodology of intelligent control. The essence of the principle becoming an essential part of our scheme lies in using fuzzy sets to address essential representation issues and afterwards make use of neural networks in order to realize all relevant computations. Furthermore we should be aware that the appliedrepresentationscheme has a direct impact on efficiency of learning by focussing its field of performance. It means that by accepting the linguistic labels (i.e. nonlinear membership functions) a preliminary nonlinear preprocessing takes place. This stage highlights some potentially fruitful andrelevantregionsof search to be explored within further detailed learning. In this way the optimization search is not solely guided internally by a prespecified universal performance index (as it happens in almost all the existing learning schemes). One has to be aware that the "universal" performance index makes the learning insensitive to the problem specificity. The concise summary of the relationships between neural networks and fuzzy set

333

techniques is displayed in Fig. 1.

Fuzzy Sets (knowledgerepresentationlevel of control)

Neurocomputations (conputational aspects of control)

Fig. 1. Relationships between fuzzy sets and neurocomputations Fuzzy sets and nuerocomputations exhibit an evident synergy in this structure. Nevertheless, they also complete separate and well-defined tasks. The possible links between them might be, of course, mutual in their nature. As already said, fuzzy sets provide an external guiding mechanism for the neural network. The network, while learning proceeds, can influence the level of knowledge repnesentation (e.g., by modifying the shapes of membership functions and changing their mutual distribution). For illustrative purposes let us focus on fuzzy controllers [6] [10] [12] and control expert systems [1] as two interesting and important constructs in control systems. Despite existing differences they share one common property, which resides within their rule-based knowledge representation scheme. Furthermore, in fuzzy control we have fuzzy predicates involved in condition and control parts of the individual rules. Usually learning mechanisms are not present there since, by default, one assumes that the entire knowledge is given explicitly. This, however, might not be true. Then if the learning mechanisms were incorporated they could significantly enhance and speed up the procedure of designing the conu-oller. In addition to that, because the rules are just generalizations of a series of facts, the use of the fuzzy predicates in all the control rules is legitimate as a suitable means to generalize over a collection of facts including some essential exceptions. To move into details let us take a look at some learning aspects and representation capabilities of fuzzy controllers and neural networks. These comments will highlight why the synergy between these two concepts is beneficial and inevitable, (i) Most of the fuzzy controllers used nowadays combine their control rules Aj^Bi, i= 1,2, ... N

334

N

by forming a union of their Cartesian products of the form u

(Aj x Bj). The inference

n= l

is based on a standard max-min composition operator [6] [10]. This way of aggregation and inference has been broadly implemented giving rise to many application-oriented versions of the fuzzy controllers. One should be aware that this design indicates a strong resemblance of the fuzzy controllers to associate memories that utilize the Hebbian-like encoding and correlational recall.' Furthermore, as studied in [6], the knowledge incorporated into the structure of the controller has to be consistent. Even the fuzzy controller can tolerate some discrepancies of the rules, starting from a certain level, these shortcomings tend to affect the performance of the controller. If an appropriate learning mechanisms were in place these inconsistencies would be easily ruled out The set-theoretic operations (like max and min) are not capable of eliminating some exceptions and inconsistent facts. The facts, even being very rare and quite meaningless, could easily override a vast number of consistent observations that have been already "coded" into the structure of the controller. (ii) In the domain of neural networks the learning capabilities of them are significant. Usually their representation features are not enhanced. The first problem arises when input variables refer to different physical variables and, what makes the entire story worse, these variables can take on numerical values from distant ranges (say 10"^ as opposed to 10*). The learning using the raw data is almost impossible from a numerical point of view. This calls for normalization and/or discretization of the input variables as a significant preprocessing phase. In the next section we will focus on a general topology of intelligent controllers enhancing the general methodological links as visualized in Fig. 1. In Section 4 we will set up a general architecture of the controller and proceed with underlying learning schemes.

3. TOPOLOGY OF INTELLIGENT CONTROLLERS: REPRESENTATION AND PROCESSING ISSUES In this section we will study a general scheme of the controller, identify its basic functional blocks and finally come up with its detailed scheme including underlying mathematical formalism. 'To make this analogy clear lei usreplacethe union (u) by summation and the minimum operator (x) by a standard product operator.

335

Let us start with a multivariable control structure comprising of several input variables (variables of the system under control) and control variables. Fuzzy sets (FS) and neurocomputations (NN) are directly involved in different parts in the diagram as shown in Fig. 2.

system variables control variables

Fig.2. Functional scheme of the controller involving Juzzy sets and neural networks Its individual blocks are described as follows. Firstly, all the systems variables are preprocessed by fuzzy sets contributing to the cognitive perspective suitable for this specific variable. [7] [8] This nonlinear transformation situated there enables us to come up with a vector of numbers in [0,1] intervals and build all the inputs of the neural network. By this mapping mechanism, all physical quantities of the variables are translated in a nonlinear manner into the corresponding dimensionless quantitates situated in [0,1]. Note that this translation provides a mechanism of nonliunear data quantization. Subsequently, the outputs of the network indicate degrees of activation of control labels of all control variables. This request calls for another conversion block (C) the role of whose is to convert any information coming from the processing block into a single numerical quantity. The more detailed structure of the controller giving a better insight into the functional blocks and their performance is shown in Fig. 3.

336

input data - pointwise defined -interval-valued —

- fuzzj sets

>

•

linguistic labels of system variables

-x->-

single-layer neural network

-^

linguistic labels of control variables

Conversion block (pointwise control representation)

- Incomplete

Fig. 3. Detailed structure of the controller It is worth noting that the topology of the network is influenced by the specification already done with regard to the representation part. The number of the layers and nodes of the neural network depends upon: (1) the number of the linguistic labels associated with the individual system variable; the number of the input nodes depends on all the linguistic labels defined for the variables. Each input node correspond to a single linguistic label. Denote their number by "n". (2) The number of the linguistic labels describing control variables uniquely determines the number of hidden nodes. Let its numbers be equal to " h". They form a hidden layer of the network. Before proceeding with learning to be completed by the network, we will briefly describe the way in which the input nodes are activated as well as highlight the manner in which thefinalnonfuzzy value of control is determined. Considering the linguistic label as uniquely described by its membership function A, where A(x) stands for degree of membership at element x, we express the level of activation of A caused by any fuzzy (or nonfuzzy) information X accordingly sup [min(A(x), X(x))] (1) where supremum is taken over all the elements of the universe of discourse in which A and X have been already defined. Note that the above expression is nothing but the possibility measure of X taken with respect to A [13]. It plays a role of a scalar index pointing out a degree to which A is "fired" (activated) by X. As already mentioned, the preprocessing block can handle various forms of informadon:

337

if X is given precisely as a single-valued numerical quantity x„, then the membership function X(x) is centered exclusively at x„, namely

X(x) = 1 ' ^~^ |0, otherwise In virtue of (1), the degree of activation of label A is equal to X = A(xJ, i.e., we obtain a grade of membership of x^ in A. if X is partially known and allocated anywhere within given lower and upper boundaries, say X = [x-, x+], then, again making use of (1), the activation level is equal to H = sup A(x) X€ [x-, X+]

see also Fig.4

Possibility 1.0

Fig. 4. Computations of possibility measure for pointwise and interval-valued input X Obviously, if x„ e [x-, x+], then this type of information implies higher activation of the corresponding node in comparison to that generated by the pointwise datum. The output layer realizes a nonfuzzy control action. For the t-th control variable, u„ it is computed by specifying a nonfuzzy representation of fuzzy sets of control (expressed by the hidden layers) and treating them as the connections from the hidden layer to the t-th node of the output layer. The operation realized at this output is expressed as ut = (utiyi + Ut2y2 + - + uthyhVCyi + y2 + - + yh) (2) where Utj are connections between the t-th output node and the j-th node of the hidden layer while y,, yj, .... y„ are the signals generated by this layer.

338

In the two particular situations expression (2) converts into: - if only a single hidden node is activated, say the j-th one, j e {l,2,...,h), yj = 1.0 and zero otherwise, then the output Ut becomes equal U, = Uj

- when all the nodes in (2) are activated, the resulting control u, becomes a mean value of the representatives, (connectives) uy h

"j=i

^.

LOGIC-BASED NEURAL NETIilORKS : LEARNING

ARCHITECTURE AND

In this section we will discuss a detailed architecture of the neural network. Essentially, it consists of two components. The two first layers are constructed by the two types of logical computing nodes (neurons). These types of neurons will be referred to as logical neurons. The last layer of the network is built with the use of a nonlinear mapping realizing (2) which is also viewed as a component carrying out a transformation from the level of the logical relationships to the physical level of the control variables. For the sake of clarity in the remainder of the section we will discuss the controller with a single output (the extensions to the multidimensional case are quite straightforward). The overall architecture of the controller is illustrated in Fig. 5 It consists of two hidden layers. The first hidden layer is composed of AND neurons while the other utilizes OR-type processing units. Essentially, these two layers encapsulate a collection of "if-then" control statements. The AND neurons summarize all the inputs (subconditions) by ANDing them. The conclusion (control) part is aggregated ORwise. The detailed formulas describing the network are summarized below. Note that the realization of the AND neurons involves the usage of the triangular norms (t- and s-norms) that result in building a s-t composition.

339 hidden2

output

input

numerical

if-then rules

transformation condition laver

conclusion layer

Fig. 5. Architecture of logic-based neural network

output-hidden layer:

'

yiUi+y2U2+...+yhUh yi+y2+...+yh

hidden^-hiddeiij layer:

yi = .S (wij t Zj) j=i

i=l,2,...,h hidden,-input layer:

Zj = jT (vji s xi) j=1.2,...,p

340

Now we will address the problem of supervised learning. This mode of learning implies the use of a collection of the training samples organized as inputs x,, x^, ..., x^ and associated outputs viewed as target values t,, tj, .... t^. The learning pertains to updates of the connections (w,v,ii) via a steepest descent method, A_connections= -a :r ^^-^ oconnections The Mean Square Error (MSE) will be used to optimize the network: N

Q= X[tk-N(xk)F where N(x^) (= y^) denotes the output (control) produced by the network and obtained for the input x^. Two schemes of learning will be analyzed: (i) reinforcement learning. [ 11] In this type of learning the logic-based part of the network will be modified while the connections ui, U2, ..., Un between the hidden and output layer remain unchanged . The connections W|j are adjusted on a basis of the gradient of Q,

. 3w

The detailed formulas read coordinatewise as Wij = Wij + a ( t - N ( x ) ) - ^ - ! ^ OWij

i = 1, 2, ..., h. j = 1,2

p, a 6 (0,1) Vij = Vij + a ( t - N ( x ) ) ^ ^ ' ' ^ ij

i=l,2,...,p,j = l,2,...,n. Furthermore,

aN(x) _ aN(x) ayi 9wij

ayi

dwij

The first derivative equals aN(x)

Ui-N(x)

ayi

a

h

where 0 = ^ yi. The second derivative can be expressed explicitly once the triangular 1=1 norms have been specified, | ^ = 5 ^ ( , S (wiiiz,)) dWij dWjj 1=1 To compute the adjustments of Vy we calculate.

341

aN(x) ^ y 9N(x) 3yi 9zi with

and 5 ^ = 5 — ( T (Vii s xi)

ii)

All the connections of the logical part of the network as well as the control values ui, U2, ..., Uh are trained. The values u; play now a role of "optimal" prototypes similar to that obtained in Adaptive Vector Quantization (AVQ), cf. [9]. The learning process itself reminds the search of an optimal partition of the control space U (in which all the control variables are situated). The resulting general learning scheme is the same as already presented for (i) with an additional update scheme describing a sequence of modifications of Uj's, Ui = U i + a ( t - N ( x ) ) ^ p ^ dui i = 1, 2, ..., h with 9N(X) ^ yj aui

5.

^

CONCLUSIONS

The paper clarifies the role of coherence between the processes of knowledge representation ad its pro.essing that is necessary to develop intelligent controllers. By changing the scope of interest and the level of precision of the linguistic labels, we can easily develop a suitable perspective required to cope with complexity of the system and specificity of the control actions. Furthermore, as it has been clarified, by changing the number of the labels we can easily switch from almost symbolic level of processing (very few labels representing a very coarse space partition) to the very fme and almost pointwiselike numerical grains of information. At this point the processing of control knowledge is

342

worked out on a numerical level with the aid of logic-oriented neural networks. Two learning scenarios are also studied. The fuzzy set interface represented by the collection of linguistic labels is advantageous from at least three major reasons, the controller can handle precise, incomplete, or noisy data pertaining to system's variables. the connections of the neural network can be easily interpreted as a qualitative conu-ol protocol involving conditional statements of the format "if state the control". These statements are generated in an automatic manner thus the scheme can be viewed as an efficient tool for automatic rule generation or adjustment of some preliminary sources of knowledge. This may allow us to avoid a serious botdeneck occurring in many knowledge-based systems and related with the tedious and error-prone knowledge acquisition processes, finally, the labels heavily influence the topology of the network determining the number of its nodes in the input as well as in the output layer. Therefore some relationships between information granularity and learning capabilities of the conU-oUer as well as its robustness can be clarified and a reasonable trade-offs established. In this manner we can talk about focussed (external) learning implied by fuzzy sets. ACKNOblLEDGMENT Support provided by the Natural Sciences and Engineering Research Council of Canada and MICRONET is gratefully acknowledged

REFERENCES [1] K.J. Astrom, J.G. Anton, K.E. Arzen, "Expert Control", Automatica, 13, 1986, 277-286. [2] B. D'Ambrosio, Qualitative Process Theory using Linguistic Variables, SpringerVeriag, New York, Beriin, 1989. [3] K. Forbus, Qualitative Process Theory Ph.D. Thesis, MIT, Cambridge, Mass., July 1984. [4] IEEE Control Systems Magazine, (Special section on neural networks and control). Vol. 8, no. 2, 1988. [5] R. Moore, Interval Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1966. [6] W. Pedrycz, Fuzzy Control and Fuzzy Systems, Research Studies Press, J. Wiley,

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343

Chichester, 1989; 2nd extended edition, 1993. [7] W. Pedrycz, "Fuzzy Set Framework for Development of Perception Perspective", Fuzzy Sets and Systems, 37, 1990, 123-137. [8] W. Pedrycz, "Selected Issues of Frame of Knowledge Representation Realized by Means of Linguistic Labels", Int. J. of Intelligent Systems, 7, 1992, 155-170. [9] W.K. Pratt, Digital Image Processing, J. Wiley, New York, 1978. [10] M. Sugeno (ed.) Indusuial Applications of Fuzzy Control, North Holland, Amsterdam, 1985. [11] R. Williams, "Reinforcement Learning in Connectionist Networks: A Mathematical Analysis", Univ. of California at San Diego, Inst, for Cognitive Science, Technical Rep. 8605, 1986. [12] L.A. Zadeh, "A Rationale for Fuzzy Control", Trans. ASME, Sec. G., 94, 1974, 3-4. [13] L.A. Zadeh, "Fuzzy Sets and Information Granularity", In: Advances in Fuzzy Set Theory and Applications (M.M. Gupta, R.K. Ragade, R.R. Yager, eds.). North Holland, Amsterdam, 1979, pp. 3-18. [14] L.A. Zadeh, "Fuzzy Sets as a Basis for a Theory of Possibility", Fuzzy Sets and Systems, 1, 1978, 3-28. [15] L.A. Zadeh, "Fuzzy Sets", Information & Control, 8, 1965, 338-353. [16] L.A. Zadeh, "The role of fuzzy logic in the management of uncertainty in expert systems". Fuzzy Sets and Systems, 11, 1983, 199-227.

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PART 4

FUZZY REASONING AND ESTIMATION METHODOLOGIES

CHAPTER 13

FUZZY PARAMETER AND STATE ESTIMATION S.G. Tzafestas(*), S. Terzakis^*) and (*)

(**)

A. N. Venetsanopoulos^**)

Intelligent Robotics and Control Unit Department of Electrical and Computer Engineering National Technical University of Athens Zografou Campus, Zografou 15773, Athens, Greece Department of Electrical Engineering University of Toronto, Toronto, Ontario, Canada M1551A4

1. INTRODUCTION The field of parameter estimation goes back to Gauss (1795) who has presented the classical least squares method in his book : "Theoria Motus Corporum Coelectiwn". Least squares was then studied by Legendre (1806) in his book: "Nouvelles Methodes pour la Determination des Orhites des Gametes" . In Gauss' own words : "The most probable value of the uknown quantities will be that in which the sum of the squares of the differences between the actually observed and the computed values multiplied by numbers that measure the deegre of precision is a minimum". Since then, an enormous effort was made by scientists and engineers to develop further the method and apply it to sophisticated situations (least squares in function spaces, least squares filtering, etc.) and applications of industrial and real-life nature.Regarding the fuzzification of the least squares problem, the authors have picked-up the work of the Hungarian mathematician Celmins [1]. He has developed an elegant but quite involved, computationally, method along with a software package (called COLSAC) that implements it. The authors present an alternative (less general) least-squares method where use is made of the concept and the algebraic properties of the so called L-R fuzzy numbers [2]. Minimizing their distance with respect to the unknown parameters is a classical minimization problem and has a small computational demand. The chapter continuous by formulating and solving the fuzzy state estimation of a particular discrete time state-space model with fuzzy disturbances and initial conditions [3-7]. Similar problems have been considered by several authors under various mathematical formulations. For example, it is worthwile to mention the work of Lee [8] where a state estimation problem is solved for a class of distributed-parameter systems with uncertain parameters. The parameters are assumed to be arbitrary time functions known to be in a closed and bounded region. The resulting estimation algorithm provides an assured accuracy and a "guaranteed" error estimator which gives an upper bound of the estimation error for any allowed variation of uncertain parameters [9]. Other works on fuzzy estimation include [10-20]. The structure of the chapter is as follows. Section 2 starts with a general discussion of the fuzzification of the classical least squares problem, proceeds to a brief review of Celmins' fuzzy least squares method, and goes on to the formulation and solution of the authors' fuzzy least squares problem. Section 3 deals with the state estimation problem of 347

S.G. Tzafestas and A.N. Venetsanopoulos (eds.) Fuzzy Reasoning in Information, Decision and Control Systems, 347-368. © 1994 Ktuwer Academic Publishers. Printed in the Netherlands.

348

fuzzy dynamic systems, including a discussion of the properties of n-dimensional fyzzy sets. All the required background concepts and properties of fuzzy real arithmetic numbers and L-R fuzzy real numbers are collected in the Appendix (Section 5). 2. FUZZY LEAST SQUARES

PARAMETER ESTIMATION

2.1 General discussion The classical least-squares estimation problem is formulated as follows: GIVEN: The data set (xj,yj), i=l,2,...,n and the model Yj=f(xj,a) where a is a parameter vector, FIND: an optimal estimate of a. SOLUTION: The solution is the value of a that minimizes the total error: n

2

where ej=Yj-yj is the ith error and Cj, i=l,2,...,n are given weighting coefficients. The optimal estimate is found by solving the equation dE/da=0 , i.e ' dE a: -r-=0 da If (xj.yj) are normally distributed and f(.) is linear, then the least squares estimate of a coincides with the most probable estimate of a, since the total error E is the negative exponent in the probability distribution of a given (xj,yj), i=l,2,...,n. Now we shall try to insert "fuzziness" in the above classical least squares problem. Fuzzines can exist either in the data of the problem or in the parameters of the model. Before proceeding to the formal derivation, let us give the following real example. A student tries to make his reading plan in order to take the exams of the "fuzzy control" subject. From his own experience he knows the following: -With "about" 2 hours/day study he solves "about" 2 problems -With "about" 3 hours/day study he solves "about" 4 problems -With "about" 4 hours/day study he solves "about" 7 problems -With "about" 5 hours/day study he solves "about" 11 problems -With "about" 6 hours/day study he solves "about" 13 problems Question : If the student devotes "about" 3.5 hours/day for his study, how many problems will solve? This problem is actually a problem of fuzzy estimation. In relation to this problem one can observe the following: (i) The student (the human) is a dynamical system that receives stimuli (inputs) and gives replies (outputs). Here both the inputs and outputs are fuzzy (they contain the word "about"). Thus we have a fuzzy system and we want to estimate its model, (ii) The human replies very easily to questions (problems) that involve fuzziness, even if he knows nothing about theories ( e.g. the least squares theory). A plaussible human answer to the above question is that: "With "about" 3.5 hours study per day the student will solve "about" (more or less) 6 problems".

349

Let us now, try ro formulate this problem with classical mathematics. The problem data are the pairs (Xj.Y^) where X j , Yj are fuzzy numbers e.g. Xj="about 2". A model that might be used has iht form y=a+bx where a and b are real-valued coefficients. The ith error is ej=yj-Yj=(aj+bXj)-Yj. Now, one has to take into account the following observations: Observation 1 : The classical numeric operations cease to have sense, e.g. how much it is 4+2 * "about 2 " ? What is the meaning of "+", " , " , " / " , etc? Observation 2 : How one can represent concepts that involve fuzziness, e.g. "about" , "almost" , " more-or-less" ? Observation 3 : The error ej in the classical problem is actually the distance between the numbers yj and Yj. Therefore we need to define the distance between fuzzy numbers, e.g. the distance between the fuzzy numbers "about 3" and "more-or-less 2". Observation 4 : The concept of differentiating with respect to fuzzy numbers is required, e.g. how one computes the derivative: dE/d("about-a") ? To overcome the above difficulties one can follow two approaches; namely (i) to reduce the problem into a classical one, (ii) to keep the fuzziness and treat it directly throughout the problem solution. When using the first approach the fuzzy problem is approximated by a classical one, the solution of which is presented as an approximation to the solution of the original fuzzy problem. In the second approach the fuzziness is not eliminated but kept throughout the solution. Thus the question arises: What is better, an exact solution to an approximate problem or an approximate solution to an exact problem? Here we shall concentrate on the second approach. 2.2 Celmins' Fuzzy Least Squares Problem A fuzzy formulation of the least squares problem was provided by the Hungarian mathematician Celmins [1]. Our purpose here is to briefly review his results. According to his notation the classical least squares problem is defined as: Minimize W=llCj^ll2 s.t. F(X+Cj^,t)=0 where F(. , .) is the model function, X+Cx is the data vector (corrected by C^), C^ is the error, t is the parameter vector, and II. II a suitable norm. In summarized form the problem is defined by the quadtuple : { W;IICxll;F;W=IICxl|2}. This method can be fuzzified in two ways: (i) When the data are fuzzy In this case instead of X we have (X,|j,jj((Cx)), and the problem becomes: Miniinize V^=il-[i^(Cy.))'^ subject to FCX+C^, t)=0 i.e we have the problem ( X; [i^iC^y, F; W=(l-ii^(C^))2 }

350 (ii) When the model is fuzzy Here, instead of the function F we have (F,[ip), and the problem becomes: Minimize W=(l-|ip)2 subject to F(X,t)=0 , i.e we have the problem { X; IIC^INO; F; |Xp W=(l-fip)2 ). Here we shall consider the first case. The second requires the solution of a fuzzy linear regression problem. Let us first study topologically the set of fuzzy numbers with conic membership functions in one or more dimensions. If "a" is a fuzzy number with conic membership function, then: |x^(x)=max(0, l-lx-al/s^) = l-min{I, lx-al/s^ =l-min(l, llx-alla) where llx-alla=lx-al/s^ is the associated norm, a is the dominant value, and Sjj is the spread. If A is a fuzzy vector then HA(x)=max(0, l-[(x-A)Tp^-l(x-A)] H = l-min{l, [(x-A)Tp^-l(x-A)] ^^} = l-min{l,IIX-A'llA) where the norm is ||X-A i|j4^ = [(x-A)Tp^-l(x-A)]-^ • with A vector of dominating values, and P ^ diagonal matrix with elements the spreads of sdj. Pictorially a fuzzy 2-dimensional vector with conic membership function is shown:

H^^ 1

Fig.l

-•

Fuzzy vector with conic membership function

351

Celmins defines the distance D(A,B) between two fuzzy vectors A and B: D(A,B)=minmax{h(X,A),h(X,B)} where h(.,.) is a metric between a classical vector X (X^ R"^) and a fuzzy n-dimensional vector A. Celmins' formulation of the fuzzy least squares is then the following: Minimize n

i=l

subject to Fi(X,+Ci, t)=0, i=l,2,...,n

llcjix, IIC; II p/( i c J| x + llcillF.)
where y* is a suitable constant. The above model is applied by Celmins to the following real problem: Find a fuzzy model of the form V=a+bQ, with a,b fuzzy quantities, which connects the "fuzzy" variables V (impact velocity of missile upon target) and Q (missile's shape). He developed a statistical software package COLSAC (General least square computer program). Celmin's method is quite mathematically involved and has a very high computational complexity. As a result, its application needs the use of the specialized statistical package ( COLSAC). In the following, a new fuzzy least squares method, developed by the authors will be presented, which possesses much less computational complexity than Celmin's method and is easily applicable to practical cases. 2.3 Formulation of a new Fuzzy Least Squares Problem This formulation is based on the use of L-R fuzzy numbers (see Appendix) and their very simple form of distance d[.,.]2, and reduces the problem to that of finding the minimum of a classical function. The fuzzy numbers of type L-R constitute a special class of fuzzy numbers very useful in estimation and other applications. Specifically, a fuzzy LR number M has a membership function of the type (see Appendix): L((m-x)/a) , x<m

{

R((x-m)/p , x>m

where m is a classical number, a and P are parameters, and L(.), R(.) are functions of special type (satisfying the conditiqns of Def A.l in the Appendix). Symbolically an L-R fuzzy number M is denoted by :

M=(m,a,P)

352

Now, let F(R) the set of all fuzzy L-R numbers that are defined on R. On the set F(R) one can define a linear structure, since the following properties hold: a) (m,a,P)+(n,Y,8)=(m+n,a+Y,P+6) p) t(m,a,3)=(tm,ta,t3) , t>0 Y) t(m,a,p)=(tm,t3,ta) , t<0 Also, on the set F(R) one can define a norm d[.,.]2 via which the distance of two fuzzy L-R numbers Xj=(xj,aj,3j) and X2=(x2,a2,f^2) can be determined: d[Xi,X2]2=(xi-X2)2+(xi-X2-(ai-a2))2+(xi-X2-(3i-|32))2 The new formulation of the fuzzy least squares problem is then the following: GIVEN: Fuzzy R-L data {X-^X{) , i=l,2,...,N where XpCxj.cj.di), Yj=(yj,ei,fi) and FIND:

a linear fuzzy parametric model Z=a+bX where a,be R or F(R) The parameters a,b so as to minimize the total error N

^^d[Z„Y-f E= i=l

where Zj=a+bXj, and d[Zi,Yj] is the ith error (distance of the L-R fuzzy numbers Zj and Yi). It is a model, that when b in Z=a-t-bX is an L-R fuzzy number, then the product bX is only approximately L-R fuzzy number [2], i.e . (m,a,P)- (n,Y,8)=(inn,mY+na,m8-i-nP). Also, if the L-R numbers used are of the special type (m,a,a), then the computational load is reduced considerably. In the following we shall treat three cases, namely; CASE A: Model Z=aH-bX , a,be R , Xe F(R) CASE B: Model Z=A-fbX , A,Xe F(R) , b e R. CASE C: Model Z=A-i-BX ,A,B,XG F(R). 2.4 Solution of Fuzzy Least Squares Problem: Case A This is the case where we have the model Z=a+bX with a and b real numbers and X fuzzy L-R variable. We must distinguish between the two cases b>0 or b<0 since the product bX is defined differently if b>0 or b<0. (I) b>0 We have a+bXi=a+b(Xi, Cj, di)=a-(-(bXi, bc;, bdj)=(a-i-bxj, bc;, bdj)

353

and so d[Zi, Y/=d[a+bXi, Yi]^=d[(a+bxj, bdj, bci),(yi, e;, f;)]^ 2

2

2

=(a+bXi-yi) +[a+bxi-yi-(bdi-ei)] +[a+bxi-yi-(bCi-fi)] The criterion E is then equal to: N

E-J{(a+bXi-yi)V[a+bXj-yi-(bCi-ei)]V[a+bXi-yi-(bdi-fi)]^} i=l

and the parameters a and b are determined by classical minimization of the real valued function E=E(a,b), i.e Parameter a: Parameter b:

^ = 9a

0

M ^ = 0 do

The resulting algebraic system of two equations in the two unknown a and b is: N

N

3Na+b5^(3xi+di-Ci)=5^(3yi+fi-ei) i=i N

i=i N

aJ(3xi+di-Ci)+b2[x^+(xi-c/+(xi-d/] i=i

i=i

N

=5^[(Xi-Ci)(yi-ei)+(Xi+di)(yi+fi)+Xiyi] i=l

This can be solved in a straightforward way. (JI) b<0 In this case a+bXi=a+b(xi, c•^, di)=a+(bxi, bdj, bc;)=(a+bxi, bdp bCj) Thus d[Zi, Yi]^=d[a+bXi, Yi]^=d[(a+bXi, bdj, bCi),(yi, C;, f;)]^ =(a+bxi-yi) +[a+bxi-yi-(bdi-ei)] +[a+bx-yi-(bCi-fi)]^ The criterion is

354

E=2_^{(a+bXi-yi)\[a+bXi-yi-(bdi-ei)]^+la+bXi-yi-(bCi-fi)]^} i=I

and so the equations aE(a,b) ^ _ _ _ ^ 0

^ and

9E(a,b) „ -3b-=0

give

3Na+bJ(3xi+di-Ci)=5!l(3yi+fi-ei) i=l N

i=l N

a2^(3xi+di-Ci)+b2[x?+(xi-Ci)V(xi-di)^] i=l

i=l

N

==2L[(xi+di)(yi-ei)+(xi-Ci)(yi+fi)+Xiyi] i=i

2.5 Solution of Fuzzy Least Squares Problem: Case B This is the case where the model under estimation is Z=A+bX, AeF(R) b ^ R , Xe F(R),where A=(a, a^, 02). Again the two distinct cases b>0 and b<0 are considered. (I) h>0 Here A+bXi=(a, a,, a2)+b(xi, C;, di)=(a+bXi, ai+bcj, a2+bdi) d[Zi, Yj^=d[A+bXi, Yi]^=d[(a+bXi, ai+bc,, a2+bdi),(yi, e„ f;)]^ 2

2

2

=(a+bxi-yi) +[a+bxi-yi-(ai+bci-ei)] +[a+bxj-yj-(a2+bdpfi)] E(a, a],a2;b)=5^d[Zi, Yj]^ i=l

Thus the parameters a, a^, 02 and b are given by the solution of the following systems of equation: ^ . 0 ^ = 0 ^ = 0 ^=0 da ' dtti ' da2 ' db

355

(II) b<0 A+bXp{a, tti, a2)+(bxi, bd^, bci)=(a+bxi, tti+bd;, a2+bci) d[Zi, Y/=d[A+bXi, Yi]^=d[(a+bxi, aj+bd;, a2+bCi),(yi, q, f;)]^ 2

2

2

=(a+bxi-yi) +[a+bXi-yi-(a]+bdi-ei)] +[a+bxi-yi-(a2+bCi-fi)] N

E(a, ai,a2;b)=^d[Zi, Yif i=l

For simplification we assume here that the fuzzy L-R number A has the form A=(a, a*, a*), i.e a2=a2=a*. Then, the algebraic system of equations that are satisfied by a,a*, and b is: For b>0

«*=[Xfi+Z^i)^2N)-b[21d,+2cij/(2N) Na 3^Xi/N+^(di-ei)/N +Na*(^5^d,+^Cij/(2N) +b2^f(Xi-Ci)\(Xi+di)Vxf]

For b<0 a=

/N

Na[35^x./N+X(di-ei)/N -Na* X ^ i + S c i ^ N )

=2^[(xi-Ci)(y,+fi)+(xi+di)(yi-ei)+Xiyi

356

2.6 Solution of Fuzzy Least Squares Problem: Case C Here, the model is Z=A+BX where A=(a, a j , 02) and B=(b,Pi,p2)Therefore; A+BXi=(a, tti, a2)+(b, Pj, |32)(xi,Ci, di)=(a, ttj, a2)+(bxi, bc^+XiPi, bdi+x^pj) =(a+bxi, ai+bCi+xjPi, a2+bdi+Xip2) d[Zi, Yi]^=d[A+BXi, YiJ^=d[(a+bXi, tti+bCi+XiPi, a2+bdi+XiP2),(yi, e„ f)f 2

2

2

=(a+bXi-yj) +[a+bXi-yi-(a,+bCi+XiPi-ei)] +[a+bXi-yi-(a2+bdi-XiP2-fi)] and the problem is to select a, a j , 02; b, Pj, P2 so as to minimize the criterion E=2d[Zj,Yi]2. The solution is found by solving the equations:

dE_o i L - o - ^ - 0 ^ - 0 da

' dtt]

' da2

' db

^-0

' dpi

^=0

' dp2

Example 2.1 We are given the following L-R fuzzy data: Y.=(ype,f,) (4, 0.6, 0.8) (3, 0.3, 0.3) (3.5, 0.35, 0.35) (2, 0.4, 0.4) (3, 0.3, 0.45) (3.5, 0.53, 0) (2.5, 0.25, 0.38) (2.5, 0.5, 0.5)

Xi=(Xi,c,di) (21, 4.2, 2.1) (15, 2.25, 2.25) (15, 1.5, 2.25) (9, 1.35, 1.35) (12, 1.2, 1.2) (18, 3.6, 1.8) (6, 0.6, 1.2) (12, 1.8, 2.4)

The problem is to fit to them the model Y=a+bX (a,be R) or the model Y=A+bX (Ae F(R), be R). Applying the procedures described in Sections 2.4 and 2.5 one can easily verify tha the results are the fuzzy straight lines: Y=1.052+0.147X Y=(1.201, 0.180, 0.180)+0.136X

3. STATE ESTIMATION OF FUZZY DYMAMIC SYSTEMS 3.1 Introductory Concepts We now turn our attention to the state estimation problem of linear dynamic systems with fuzzy initial states and fuzzy noises. Our goal is to present an algorithm for the estimation of the fuzzy state of the system assuming known dynamical equations and a sequence of output observations. The material of the present section is based on [3-7J. To

357

formulate the theory we need to define the concepts of N-dimensional fuzzy sets and Ndimensional Gaussian fuzzy sets. We know that a fuzzy set defines on the reference hyperset R a straight line section the points of which loose, towards its ends, the property to belong to it. Analogously an ndimensional fuzzy set defines on the hyperset R" a closed volume the points of which loose the propety to belong to it towards the boundary surface. Definition 3.1 Analogously to the 1-dimensional fuzzy sets, A and B are said to be n-dimensional (n-D) fuzzy sets if A-{(x,^lA(x)) U e R " , ^^{x): R"->[0,1]} B-{(x,Mx))|xeR",HB(x):R"-^[0,l]} where |iyi^(x) and Hg(x) are n-dimensional membership functions (see Ch. 1 , Def. 2.1). Some useful operations of n-dimensional fuzzy sets (fuzzy vectors) are the following: PI. Vector Sum of n-D fuzzy sets If A and B are n-D fuzzy sets, then their sum C=AeB={(x,|j.c(x)) I x€R" , M.c(x): R " - ^ [ 0 , 1 ] } where Hc(x)=sup{jiA(x-y)A iXfiCy)} y

is also n-D fuzzy set. This sum is a generalisation of the vector sum of two classical n-D vectors. P2. Crisp translation of an n-D fuzzy set A in the direction of a crisp vector y Geometrically this means the translation of the "volume", defined by the fuzzy set A, by the quantity y. The translated fuzzy set A has exactly the same shape with A, i.e with HA'(x)=^A(x-y). P of the n-D fuzzy set A

A'={(X,^A(X)) | X 6 R V A ' ( X ) : R " ^ [ 0 , 1 ] }

P3. Direct

image

under

PA'={(X,^PA(X))

P4. Inverse

image

I xeRVpA.(x):R"-^[0,l]} with

under

Q"^A'-{(X,HQ->A'(X))

Definition

I xe

Q of the n-D fuzzy R",JXQ-IA(X):R"-^[0,1]}

^PA'(X)=^IA(P~'X).

set with

A

^1Q->A(X)=|IA(QX).

3.2

A n-D Gaussian fuzzy set A with center ue Rn , variance Q and height a is defined as : A={(X, ^A(X)) I xe R " , ^ A ( X ) : R " - ^ [ 0 , 1 ] }

where

358

^^^^'^=1"

^g-(x-u)V'(x-u)^^

and is symtx)lised by the triad: A=(a, u, Q). The operations of n-D Gaussian fuzzy sets follow as special cases of the operations of arbitrary n-D fuzzy sets. Thus, let A=(a, u i, Qi) and B=(P, U2,Q2). Then GI. Translation of A by the vector v A'=(a, Uj-v, Qi) G2. Sum of Aand B In general, the sum A©B of two n-D Gaussian fuzzy sets is not Gaussian unless they are linearly dependent, i.e A=niB, me R. However A©B can be approximated by the following n-D Gaussian fuzzy set: ( - ' Qi Q2 A©B= min(a,|3), u^+Uj , — + - ; V '^ ' ^ a 1-a where a a parameter wih 0
359

Before going to the fuzzy state estimation problem, we recall that a classical n-D vector can take each time as value only one point of R", and thus it covers all R". For convenience the differences of classical and fuzzy variables and vectors are summarized in the following table: Variable Value Classical (crisp) point of R Fuzzy Section of R Classical vector Point of R" Fuzzy vector Subsetof points of R" 3.2 Fuzzy State Estimation Problem Consider a discrete linear dynamic system with state equation: x(k+1 )=Ax(k)+Bu(k)+Gw(k), x(0)=x(to) and output equation: z(k)=Cx(k)+v(k) where the initial condition ^{IQ) is a n-D fuzzy set on the reference hyperset R" , the input u(k) is exactly known, the input disturbance w(k) ia a m-D fuzzy set on R™, and the output noise v(k) is a p-D fuzzy set on RP. The matrices A,B ,G and C are all known and real-valued and have appropriate dimensions for the operations to have sense. The fuzzy state estimation problem is the following: GIVEN: (i) A system described by the above state-output eaquations (ii) The set of fuzzy control signals U={u(0),u(l),...,u(k-1)) (iii) The set of fuzzy ouput-signals Z={2(l),z(2),...,z(k)} FIND: An estimate of the fuzzy set of the state x at the time instant k, i.e an estimate of x(k). Problem Solution Here the term "estimation" means the determination (estimation) of all parameters of the fuzzy set including its membership function. The algorithm to be described makes use of the operations described in Sec.3.1 and involves two steps. Step I: Let x(k-l/k-l) be the estimate of x(k-l) on the basis of the data up to time k-1. Then the one-step ahead predicted state estimate x(k/k-l) is given by : x(k/k-1)=Ax(k- 1/k-1 )-t-Bu(k-1 )-i-Gw(k-1) Clearly, this ives the set of all posible states x(k/k-l) which is reachable from x(k-l/k-l) through the given dynamics of the system. Step 2: This step computes the corrections to x(kA:-l) that must be made in the light of the information transferred by the new output measurement z(k). More specifically: We solve the output equation z(k)=Cx(k)-i-v(k) with respect to x(k), i.e x(k)=C-1 [z(k)-v(k)]=-C-1 [v(k)+(-z(k))J. Now, the estimate x(k/k) of x(k) on the basis of data up to time k, is obtained as the section of the quantities: x(k/k-l) and-C-l[v(k)+(-z(k))]. This is so, since the estimate must belong to both of these fuzzy sets.

360 Therefore x(k/k)=x(k/k-l)n-C '[vtk]©(-z(k))] To summarize the overall estimation algorithm is expressed by the equations: x(k/k-l)=Ax(k-l/k-l)eBu(k-l)©Gw(k-l) x(k/k)=x(k/k-l)n-C"^[v(k)e(-z(k))] with initial condition x(0/0)=x(0)=x(tQ). We now have to determine the membership functions M'x(k/k-l) ^"^ M'xfk/k) °^ x(k/k-l) and x(k/k). To this end, we start from the first equation of the algorithm: x(k/k-l)=Ax(k-l/k-l)®Bu(k-l)®Gw(k-l) This equation involves: a) A product of a matrix by a n-D fuzzy set, i.e Ax(k-l/k-l) which, according to the results of Sec. 3.1 (Property P3), gives: ^Ax(k--i/k-i)(x)=Ux(k-i/k-i)(A~ x) b) The product Gw(k-l) of the matrix G by the m-D fuzzy set w(k-l). Thus I^Gw(k-i)(x)=^w(k-i)(G'' x) c) Translation of the n-D fuzzy set Ax(k-l/k-l) by the crisp vector Bu(k-l). The result is: M•Ax(k-l/k-])®Bu(k-l)('')=^^Ax(k-l/k-l)(''-Bu('<^-l)) =l^x(k-i/k-i)(A"'x-Bu(k-l)) d)Sum of the n-D fuzzy vectors Ax(k-l/k-l)©Bu(k-l) and Gw(k-l), i.e I^Ax(k-l/k-l)®Bu(k-l)®Gw(k-l)^'')

=sup(M-Ax(k-i/k-i)®Bu(k-i/x-q)A^w(k-i)(G"'q)} =sup{^^(k_]/k-i)[A"'x-Bu(k-l)-q]A^^(k_,)(G'^q)} Now, consider the second equation x(k/k)=x(k/k-l)n-C'nv[k]®(-z(k))] This equation involves the following:

361

a) Translation of the p-D fuzzy vector v(k) by the crisp vector -z(k), i.e M-v(k)®(-z(k))(x)=^v(k)(x-(-z(k)))=^^(k)(x+z(k)) b) Multiplication of the crisp matrix -C"^ by the p-D fuzzy vector v(k)©(-z(k)), i.e M-c-'[v(k)ffi(-z(k))](x)=^^v(k)f-C~'x^-z(k)] Thus finally l^x(k/k)(x)=|ix(kA-l)'^^^v(k)f-C"'x+z(k)]

4.CONCLUSION In this chapter we have studied the fuzzy least-squares parameter estimation problem and presented two methods. The first one is due to Celmins and the second to the present authors. The first seems to be computationally very demanding, whereas the second method is applicable only to the class of L-R fuzzy numbers but it has computational efficiency similar to the classical least-squares method. Then our attention was focused to the state estimation problem of systems subject to fuzzy disturbances and fuzzy initial conditions. An estimation algorithm was presented which has high theoretical value but is practically limited by the high computational demand. The area of fuzzy parameter and state estimation seems to be quite open and provides challenging theoretical and practical problems.

5. APPENDIX: A Review of Fuzzy Algebra 5.1 Fuzzy Real Numbers Let R be the set of real numbers and ne R a given real number. From the real number n one can construct a fuzzy real number N as a fuzzy set (with particular properties) that covers (the classical real) number n. When the fuzziness of N is removed it reduces exactly to n [2]. Definition A.l: N is a fuzzy real number if and only if: (i) it is a fuzzy subset of the set R of real numbers, (ii) its membership function Hj*j(x) has the following properties; a. ^^(x) is a continuous function b. Vx6 (-°o,c] , IIN(X)==0 c. |IN(X) is strictly increasing in [c,a]

362

d- Vxe[a,bJ , |iis,(x)=l e. ^N(X) 'S strictly decreasing in [b,d] f. Vxe [d,+oo), |aN(x)=0 From this definition it follows that N is a fuzzy real number if and only if N is a convex and n normal fuzzy subset of R. Some examples of fuzzy real nunbers are shown in Fig. A. 1

i L

H(x)

H(x) 1

i

c

a

l\ 1 \ 1 \

1 4 b (b)

v^ d

10 (c)

Fig. A.l: Three fuzzy real numbers (a) "about 0" , (b) "nearly 4" , (c) exactly 10 From Fig. A.l we observe the following: (i) Fuzzy 0: "about 0", here a=b, shape nearly triangular (ii) Fuzzy 4: "nearly 4" here a^tb, shape nearly trapezoidal (iii) Classical 10: "exactly 10", here a=b=c=d, shape like a spike The operations of fuzzy numbers are extensions of the respective operations of classical numbers and are defined as follows [2].

Definition

A.2: Let M={(x, [i^M) I xe R}, N={(y, ^tN(y))' ye R}

fuzzy real numbers. Then the following operations are defined: 01) Fuzzy addition M4-N={(z, maxmin (|iM(x),|j.^,(y)) | Vx,y,z:z=x+y} 02) Fuzzy

subtraction M-N={(z, maxmin (|if4(x).M-N(y)) I Vx,y,z:z=x-y}

363

03) Fuzzy

opposite -N={(z,^N(y))|z=-y}

04) Fuzzy

multiplication MxN={(z, maxmin (|iM(x),(iN(y)) I Vx,y,z:z=xy}

05) Fuzzy

inverse i/N={(z,tiN(y))lz-i/y}

06) Fuzzy

division M/N={(z, maxmin ()iM(x),)iN(y)) I Vx,y,z:z=x/y}

An example is shown in Fig. A.2

Fig. A.2

"about 2" x "about 5"="niore or less 10"

It is obvious from the above definitions (and from Fig. A.2) that the operations of the fuzzy numbers have a convolutional structure, and so they are computationally involved. For example, if Supp(N), and Supp(M)=m then Supp(MxN)=mxn. It is therefore necessary to develop a special type of fuzzy numbers, the operations of which have computational complexity equivalent to that of classical operations. Such a class of fuzzy numbers are the so-called L-R fuzzy real numbers anf are examined below. 5.2 L-R Fuzzy Real Numbers Let y(x) be a real funcdon y(x):R—>[0,1] that possesses the properties of |J.]sj(x) of definidon A.l, and can be considered the membership function of a fuzzy real number. Examples of such a function y(x) are:

364

y(x)=max(o, 1-i x | )

y(x)=max(0,l~x )

y(x)=exp(-1XI)

y(x)=exp(-x )

y(x)=l/(l+|x|)

y(x)=l/(l+x^)

Definition A.3: Let L(x) and R(x) be two functions satisfying the conditions of Def. A.l. Then as L-R fuzzy number M around the classical number m, is defined a fuzzy number M with membership function: L((m-x)/a), x<m ^IM(X)=-^

R((x-m)/p), x>m where m is the modal value and a, P, are spread parametes. Since the L-R fuzzy number is fully determined by the triad of parameters m, a and p it is symbolised by: M=(m, a, P). Example

A.l

Let L(x)=l/(l+x2) and R(x)=l/(1+ | x |) with a=2 , P=3 and m=5. Then, the fuzzy number "nearly 5" has the membership function: L((5-x)/2)=l/[l+((5-x)/2) ] . x<5 ^5M=

{

R((x-5)/3)=l/[l+ I (x-5)/3 I ] , x>5

which is pictorially shown in Fig. A.3.

0

5

X

Fig.A.3 R-L fuzzy number "nearly 5" As shown by the following Theorem A.l, the operations of fuzzy L-R numbers reduce to numerical operations of standard numbers [2].

365

Theorem A.l Let two L-R fuzzy numbers M=(m, a, p) and N=(n, y, 6). Then one can show [2] the following: / . Fuzzy addition M+N=(m, a, P)+(m, y, 5)=(m+n, a+y, P+5) 2. Fuzzy

opposite -M=-(m, a, P)=(-m, a, P)

R-L numbers

3. Fuzzy subtraction The subtraction has sense only between L-R and R-L , L-L and L-L , R-R and R-R numbers (not between R-L and R-L numbers). M-N=M-i-(-N) 4. Fuzzy

inverse l/M=(l/m, p/m^, «/m^)

R-L number

5. Fuzzy multiplication Three cases are distinguished: Case A: If m>0 and n>0, then MN=(m, a, P)(n, y, 5)=(mn, na-m5, np-my) =-[(-(m, a, P))(n, y, 5)] Case B: If m<0 and n>0, then MN=(m, a, P)(n, y, 6)=(mn, my-t-na, m5-l-np) Case C: If m<0 and n<0, then MN=(m, a, p)(n, y, 8)=(mn, - n p - m 5 , - n a - m y ) —[(-(m, a, p))(-(n, y, 5))] 6. Fuzzy

division M/N=M-(1/N)=(m, a, p)(n, y, 5)=(mn, (m&+na)/n^ (myfnPVn^)

We now proceed to extend the operators MIN and MAX in the fuzzy algebra. In the classical algebra these operators are defined axiomatically as follows: Let a,Pe R. Then by definition:

366

1. a>(3 or a^p or a

p if a - p > 0 3. a

max(in,N)

kl

k2

k3

Fig.A.4 Two fuzzy numbers and their maximum Clearly, this cannot be answered through the axioms of classical arithmetic, and so a new ordering relation is defined as follows [2]: Definition A.4 Let M={(x,

^M(X))

I xe R} and N-{(y, ^N(y) l y e R )

two fuzzy real numbers. Then Max(M,N)={(z,maxmin(jj,^j(x), [y^iy)) \ z=max(x,y)} In the special case where two fuzzy numbers M and N have only three intersection points k j , k2 anfk3, then Max(M,N)={(z, where

HMAX(Z))

I ze R}

367

I

Hj^z),ze(-oo,kiMk3,+oo)

|jfj;z),z€[ki,k3]

A pictorial example of Max(M,N) in this special case is shown in Fig.A.4.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

A. Celmins, Least-Squares Model Fitting to Fuzzy Linear Data, Fuzzy Sets and Systems , Vol.22, pp. 245-269, 1987. D. Dubois and H. Prade, Operations on Fuzzy Numbers, Int. J. Systems Sci., Vol. 9, No.6, pp. 613-626, 1978. H. Sira -Ramirez, Fuzzy State Estimation in Linear Dynamic Systems, Proc. IEEE Conf. on Decision and Control, Vol.2, pp. 380-382, 1980. H. Sira -Ramirez, Evolution of Fuzzy Sets in Linear Dynamic Systems, Proc. 1979 Int. Conf. on Cybernetics and Society, Denver, Colorado, Oct. 1979. D.P. Bertsekas, Control of Uncertain Systems with a Set-Membership Description of the Uncertainty, MIT Electr. Syst. Lab. Rept ESL-R-447, Cambridge, June 1971. F.C. Schweppe, Recuirsive State Estimation: Unknown but Bounded Errors and Systems Inputs, IEEE Trans. Auto. Contr. Vol. AC-13, No.l, pp. 22-28, 1968 H. Witsenhausen, Set of Possible States for Linear Systems Given Perturbed Observations, IEEE Trans. Auto. Contr. Vol. AC-13, No.5 pp. 556- 558, 1968. K.Lee, Modelling and Estimation of Distributed Systems with Uncertain Parameters, In: Distributed Parameter Systems: Modelling and Identification (A. Ruberti, Ed.),pp. 325-334 1978. S.S.L. Chang, Control and Estimation of Fuzzy Systems, Proc. IEEE Decision and Control Conf, pp. 313-318, 1974. P. Diamond, Higher-Level Fuzzy Numbers Arising from Fuzzy Regression Models, Fuzzy Sets and Systems, Vol. 36, pp. 265-275, 1990. IFAC Report,, Round Table Discussion on the Estimation and Control in Fuzzy Environments, Automatica, Vol.11, pp. 209-212, 1975. S. Joszef, On the Effect of Linear Data Transformations in Possibilistic Fuzzy Linear Regression, Fuzzy Sets and Systems, Vol.45, pp. 185-188, 1992. H. Tanaka and H. Ishibuchi, Identification of Possibilistic Linear Systems by Quadratic Membership Functions of Fuzzy Parameters, Fuzzy Sets and Systems, Vol.41, pp. 145-160, 1991. . A. Bardossy, Note on Fuzzy Regression, Fuzzy Sets and Systems, Vol.37, pp. 6575, 1990. K.Jajuga, Linear Fuzzy Regression, Fuzzy Sets and Systems, Vol.20, pp. 343-353, 1986. H.Tanaka and J. Watanabe, Possibilistic Linear Systems and their Applications to the Linear Regres.sion Model, Fuzzy Sets and Systems, Vol.27 pp.275-289, 1988. H.Tanaka, S. Uejima and K. Asai, Linear Regression Analysis with Fuzzy Model, IEEE Trans. Syst. Man Cybern., Vol.5 SMC-12, No.6, 1982.

368

18. 19. 20.

W. Zhen-Yuan and L. Shou-Mei, Fuzzy Linear Regression; Analysis of Fuzzy Valued Variables, Fuzzy Sets and Systems, Vol.36, pp. 125-136, 1990. S.-Q. Chen, Analysis of Multiple Fuzzy Regression, Fuzzy Sets and Systems,l9^8. D,A.Savic and W. Perdrycz, Evaluation of Fuzzy Linear Regression Models, Fuzzy Sets and Systems, Vol.39, pp. .51-63, 1991.

CHAPTER 14

A FUZZY REASONING METHODOLOGY FOR RULE-BASED SYSTEMS Shyi-Ming Chen Department of Computer and Information Science National Chiao Tung University Hsinchu, Taiwan, R.O.C.

Abstract In this paper, vvc present a methodology to deal with fuzzy reasoning based on the matching function S. The singlc-input-singlc-output (SISO) fuzzy reasoning scheme and the multi-input-single-output (MISO) fuzzy reasoning scheme are discussed in details.

1. Introduction

Much knowledge residing in the knowledge base of an expert system is fuzzy rather precise. Thus, a powerful expert system must have the capability to deal with fuzzy reasoning. Fuzzy reasoning can be viewed as a process to infer a possibly imprecise conclusion from a set of possibly imprecise premises [6, p. 320]. Many fuzzy reasoning methods have been proposed, such as [1], |2], [3], [4], [5], [6, p. 29l-341|, [9], [10], [12], In most of the above references, the following single-input-single-output (SISO) fuzzy reasoning scheme is discussed: Rule l : l F X i s A i T H E N Y i s B i Rule 2: IF X is Aj THEN Y is B2 •

(1)

Rule p: IF X is Ap THEN Y is Bp Fact: X is Ao Consequence: Y is Z:^ where X and Y are linguistic variables [15], and Ao, Ai AT, ..., Ap, BQ, Bi, B2,..., and Bp are fuzzy terms (linguistic terms), such as "slow", "medium", "fast", etc. The fuzzy terms usually are represented by fuzzy sets [13]. Roughly speaking, a fuzzy set is a class with fuzzy boundaries. Let U be the universe of discourse (i.e., a collection of objects), f/={«], H2> •••> "«}• A fuzzy set A of L' is a set of ordered pairs {(i/j,/4(HI)), (H2./4("2)), •••. ("/i,//i("«))}. where/4, Z^: U -* [0, 1], is the membership function,/^(w,) 369 S,G. Tzafestas and A.N. Venetsanopoulos (eds.) Fuzzy Reasoning in Information, Decision and Control Systems, 369-386. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.

370

indicates the grade of membership of «, in A, and 1 s / s /;. Let A and B be two fuzzy sets of the universe of discourse U, U={iii, 112, •••, u,,}, and lei/A and//? be the membership functions of the fu/./y sets A and B, respectively, whereZ^: U—*\0, !],/«: U—*10, 1], A={{iti,J\(iti)) I u,E.U}, and B={{Ui,/ij(Ui)) I iiiizU}. The fuzzy sets A and /^ arc called equal (i.e., A=B) if V iiitiU,fA(iii)=fij{ii,). The union operation between the (uzzy sets A and B is defined by

/ I u ii = {(«, ,/^ufl("/)) i^uflC,) = A^'^-^C/AC',). /«(«,)), «, et/}. Let U be the universe of discourse, U =|0, 100], and "speed" be a linguistic variable [15], and let "slow", "medium", and "fast" be three fuzzy terms. According to [9], these fuzzy terms can be characterized as fuzzy sets whose membership function curves are shown in Fig. 1. Tspeed

40 55 70 Speed Fig. 1. A diagrammatic representation of fuzzy speed. Consider the fuzzy reasoning scheme in (1), where A], AT, .., and Ap are fuzzy sets of the universe of discourse U, U={ui, 112,..., ««}, and B\, B2 Bp arc fuzzy sets of the universe of discourse of V, V={vi, VT, •., v,„}. For a given system's observation fact "X is Ao", where Ao is a fuzzy set of (J, U-{t4i, HT, ..., «„}, and the membership function of Ao ^^fAi),/An'- f^—*\0, 1|, the system can perform inference by either compositional rule of inference (CRl) [14] on each of the rules or by approximate analogical reasoning (AAR) [ 12) on selected rules from the knowledge base to reach a conclusion "Y is /Jo", where Bo is a fuzzy set of the universe of discourse V, V={vi, VT, .., v',„}. A comparison of CRI based fuzzy reasoning methods has been discussed in [10] by Mizumolo and Zimmcrmann. The approximate analogical reasoning (AAR) schema is based on similarity measures. In [16], Zwick el al. have made a comparative analysis of 19 kinds of similarity measures between fuzzy sets. They pointed out that some similarity measures (such as S4, c(»,Ao^ and A.) ha\'c relatively good performance, and the Euclidean distance measure was found not to be a gtxxl measure. However, in [12], Turksen ct al. pointed out that the conclusion in [16] may not applicable to all cases and suggested that users should experiment with different similarity measures before deciding on an appropriate measure for their t)wn use. In this paper, we present a new methodology for pcrfontung fuzzy reasoning based on the matching function S [3], [7]. The proposed methcxl belongs to the AAR scheme. The single-input-singlc-output (SISO) fuzzy reasoning scheme and the multi-input-singleoutput (MISO) fuzzy reasoning scheme arc discussed in details.

371

2. A Fuzzy Reasoning Methodology In [7], Ke el al. presented a matching function ^S' to measure the degree of similarity between two vectors. Let a and h be two vectors in R", where i? is a set of real numbers between zero and one, i.e.. a = (a^,

02, .... a,J,

(7, £

h = (/;,, 1)2, ••; h„),

[0, 1],

1S/SH,

h, E [0, 1],

1 s (" s n.

Then, there is a matching function S [3], [7],

S(a^ I)

j?_ _

(2)

Max(a * a, /; • /;) where S{a, ft)e [0, 1], which can be considered as the similarity measurement between the vectors a and h . The value of S{a, h) indicates the degree of similarity between a and h . The larger the value of S(a, h), the more the similarity between the vectors a and b. Because a* h ^ I allftlcos0, " 0 " is the angle between a and h , and because cosO = 1, (2) can be rewritten into - S(a, h)

Iall/71cos0 _ - Max(\a\\a\, \b\\b\)

(3)

The relationship between the vectors a and b can be distinguished into the following three cases: Case 1: If ff -b

(i.e., a and b are identical vectors), then S{a, b)=\.

Case 2: If Ic/I > l/?l, then S(a, b) =

\a\\b\co?.d \h\co&B lallal

\a\\b\cos6 Case 3: If lal < 161 , then .S'(^, 7)) = \b\\b\

\a\ lalcos^ \b\

Therefore, we can see that 1) If rt and h are identical vectors, then S{a, b) = \. 2) The larger the length difference between a and /;, the less the similarity between a and b . If \a\ » \T?\ or 1^1 » ki\, then Si'i, h) ^ 0.

372 3) The larger the angle 6 , Ihc less the similarity between a and b . U 6 -*

n,

then S(a, ^) a 0, where n is an even number. Based on the matching function S, we can measure the degree of similarity between fuzzy sets. Let U be the universe of discourse, U={ui, ti2,..., ««}. and let A and B be two fuzzy sets of t/, i.e., A - {(//,, rti). ("2- «2)> •••. (««, «/•)}

where fl/ fc |0, 1],fc,e [0, 1], and 1 s / s //. By using the vector representation method, the fuzzy sets A and Bean be represented by the vectors A and B, respectively, where

Based on the similarity function S, the degree of similarity between the fuzzy sets A and B can b)e measured,

I»5 .S'(A, B)

Max{A* A,

B'B)

where S{A, B) 6= fO, 1]. The larger the value of S(A, B), the more the similarity between the fuzzy sets A and B. Let's consider the following generalized modus ponens (GMP): Rule:IFXis/i| THEN Y is Bi * Fact: X is A^ — I

Consequence: Y is fi] where X and Y are linguistic variables, Aj and A\ are fuzzy sets of the universe of discourse U, U-{n\, 112, .., «„}, and Bj and Bi are fuzzy sets of the universe of discourse « * V, V={v\, VT, ..., v,„}. Assume that the membership function of the fuzzy sets A] , Ai, Bj, « , arc / . , / , / . , and / , respectively, where / . : U-*\{\ 1], / : U-^\0, 1), f . : V-^|0, 1], /' "i

: V-*[0, 1], and the I'uzzv sets Aj', Ai, and ZJ,* have the followmg

' ^1

forms: AI = {OIU -fl), (U2, XT), ..., («„, X,,)} -41 :^ {(Ml, Vl), (U2, Vl), - , (««, .Vn)} A = { ( ^ 1 . -1)' (*'2, ^2)

(V,„, -y„)}

373

where J:,£ [0, 1], ViG [0, 1], zjE: [0, 1], 1 s / s H and \& j ^m. Let Aj and A^ be the * vector representation of the fuzzy sets A^ and Ai, respectively, where Ai =(.ri, .^2, ..., x„) A\ =(.Vl, V2, ..., ,V„). Then, based on the matching function S, the degree of similarity between the fuzzy sets A, and A\ can be measured, ~^ — S(A,. A,) =

A •A - ^ ^ ^ Max{Ai • Ai, A, • A,)

where 5( A;, A,) E [0, 1]. Let 5(A|, Aj )= /:, and let the deduced consequence of the rule is "Y is 5, ", where the membership function of the fuzzy set D is as follows:

where "*" is a multiplication operator, i.e., fi,' = {(V,, k*Z{), (V2, /:*Z2)> ..., (V,„, ^ * Z j } .

* * *" It IS obvious that if Aj and Aj are identical fuzzy sets (i.e., A\ =Ai ), then S(A\, Aj )=1 * and Z^ is equal to/?]. Consider the following single-input-single-output (SISO) fuzzy reasoning scheme: Rule 1: IF X is Ai THEN Y is S, Rule 2: IF X is Ao THEN Y is B2

Rule p: IF X is Ap THEN Y is Bp Fact: X is Ao Con.scquence: Y is BQ where Ao, Aj, A2, ...,and A„ are fuzzy sets of the universe of discourse U, (/={«], 112,..., «„}, and Bo. ^1-^2. •••, ana ^p are fuzzy sets of the universe of discourse V, K={V],V2, ..., v,„}. Assume that A, = {(Hi, Jr,l), (U2, Xfi), .... (M,„ X,,,)}

Bj - {(v\, yjx), (V2, ypj,...,

(v,„, Vym)}

374

where \ ^ i ^ p and 1 s y s p . By using Ihc vector representation method, the fuzzy set A, can be represented by the vector A^ ,0 ^i s p, where

\) = vol'

''^('2'

•••'

-^Oii)

Ai ~ \ - ^ 2 l '

"^22' •*•'

^2n)

Based on the previous discussion, we can get the following results: S{\^,

A\)=ki ^> the deduced consequence of rule R[ is "Y is B^ ", where «i ={(vi, A:,*yii), (V2, *i*.Vi2), •.•, (Vm, ^i*.ViJ}.

i'(A)' ^ ) =^^2 =^ the deduced consequence of rule ^^2 '^ "Y is ft ", where / ^ = {(*'l. ^ 2 * V 2 l ) . (l'2, ^ 2 * y 2 2 ) . - . (Vffl, ^2*y2;«)}-

.S'( A)' ^p)=^p =* "ic deduced consequence of rule Rp is "Y is B^", where «^ = {(vi, Vvpi), (V2, kp*yp2)...., (v,„, V V " ) } where ^,fci [0, 1] and \-^i ^p, and the deduced consequence of the above SISO fuzzy reasoning scheme is "Y is HQ", where

and " U" is the union operator of fuzzy sets. That is, /^()= { ( V l , Z i ) , (V2, Z2), ..., (V„„ /,„,)}

where :.2=Ma.\(ki*yi2,

z,fc |0, U. iind 1 ^/ s « .

^2*.V22. •••- ^p*.V/,2)

375 Let's consider the following generalized modus ponens: Rule: IFX is Ai AND ... AND X/is A/THEN Y is B Facts: Xi is A^ AND ... AND X;is J\ ?

Consequence: Y is B * where Xi,..., X/, and Y are linguistic variables. Ai andAj are fuzzy sets o\ the universe of discourse (J, U={u\, 112, .., ««}, ..., v4/and Aj are fuzzy sets of the universe of discourse 7', T={ti, /2, •••,'/}, and Band If are fuzzy sets of the universe ofdiscour.se V, V={i'i, V2,..., v,„}, where ^1 = { ( " l , . T n ) . ('<2,-^12)> - ,

("mXln)}

^1 = { ( « ! , Vll), («2,.Vl2). - , ("/h.Vl«)}

A = { ( / ] , Xn), (/2. -^C)- -M (//, Xif)}

B={{VuZi),{V2,Z2),-,{V,n,Z,n)y

By using the vector representation method, the fuzzy sets A^ , A,,..., A^ , and A/ can «•

*•

be represented by the vectors A, , Aj ,..., A/ , and A/ , respectively, where A, ^'

V I h -fn'

•••'

-^ui)

=(.V|l, .V12. —

.Vl„)

A, = ( . t „ , .V,2, ....

X,j)

A =(.v/i. .v/2. - . y//)Based on the matching function .V, we can get the following results: S{AI

Ai)=ki

.V(A,*,

Ai)=ki

376

where ki^E [0, Ij, and 1 s / s / . Letc = Min{k\, ...,kiy, Ihe deduced consequence ol the rule is "Y is B*", where

(i.e., /g-(v^) = c*/H(V ), where / = 1 , 2 , . . . , m). It is obvious that if Aj and A\ are * identical tuzzy sets,.... and Af and Ai arc identical luzzy sets, then FT is equal to B. Furthermore, consider the following multi-input-single-output (MISO) fuzzy reasoning scheme: Rule 1: IFXi is An AND... AND X/is Aw THEN Y is Bj Rule 2: IPX, is A21 AND ... AND X/is A2,THEN Y isZ?2 •

(4)

Rulep: IFXi is A^i AND ... AND X/is A,,/THEN Y is fip Facts; Xj is Aoi AND ... AND X,is AQ/, Consequence: Y is Z^ where Aoi, An. A21,..., and A„i are fuzzy sets of the universe of discourse f/, U={u\, 112, ..., «„},..., At)/, Ai/,A2/,..., ana Ap/are fuzzy sets of the universe of discourse T, 7={/i, h, ..., If}, dnd Bo,B\, B2, ••, and Bi are {\i7.z\ sets of the universe of discourse V,V={vi, V2,..., v,„},i.e., ^401 = {(«1, -^Ol)> ("2. -«'02). •••. («H. -^Ow)} All = {(«!, .TH), («2.Jri2), - , iUn,Xln)} A21 - {(Ml, .^21), (H2. -^22). • -. («H, •*2fl)}

Api = {(Ml, .r,,i), (//2, .^,,2), - , («,„ V)5'

Av = {('1, Vol), ('2, .V02), •-, ('/, VO/)}

Au = {(?i,.viiX ('2..V12) C/.vi^} A2/ = {('], V21), (f2,.V22). -. C/.v:^}

'4,'/ = {('h Vpl), (/2, .V,>2), •-, ('/ Vp/)}

377 ^ 1 = { ( V i , J i l ) , (V2,2i2), . . • , ( v , „ , . r i J } B2 ={(Vi, :2i), (V2, :22). •••> (Vm. Jf2/n)}

B/7 ={(Vl, Zpl), (V2, Zfi), ..., (V,„, Xp,^}

By using the vector representation method, the fuzzy sets AQI, A n , A21, •••, A ^ j , . . . , AQ/. Ai/, A2/,..., and Api can be represented by the vectors A Q ] , A J J , / ^ J , ..., A ^ j , ..., AQ/ , A]/, A2/,..., and Apj, respectively, where

Al

°° \-^ll' •*'l2' •••' ^\n)

Aj I =\-^21> -^22' • •> -''2n/

'^pi

~ v ' ^ / ' i ' -^p-'

•••' •*/"'/

Av

=CvOb yo2, - ,

Vo/)

'^/

=(,v2i. .V22. • - .

yif)

(>V.- .v,„ ..., y,,/). Let

^'(Aoi, A^,) =

fc^,,...,.V(Ao,,

Ap,)'kp,

378 and let ci=Miiiku,-,ku)

Cp=Minikpu...,kpi). Then, based on the previous discussion, we can get the following results: the deduced consequence of rule Ri is "Y is B, ", where S, = {(v,, r]*sii), (V2, ci*;i2),..., (v^,
the deduced consequence ol rule R2 is "Y is B,", where A

= {(Vl, <.-2*J2l), (V2, C2*;22), •••• {Vm, C2*Z2,rd};

* the deduced consequence of rule Rp is "Y is B„", where Bp = {(Vi, Cp*Zp\), (V2, Cp*Z,i2), - . (I'm C ^ % « ) } -

In this case, the deduced consequence of the above MISO fuzzy reasoning scheme is "Y is Bo", where £^ =^"' U f t U ...UB^p

and " U" is the union operator of the fuzzy sets, i.e., lh={{Vu

S]), (V2, S2), .... (V,„, A j }

where si=Majiici*zn,C2*Z2\, ...,Cp%i) .V2 = MaJ((C]*Zi2< '•2*^22, •••. f'>*2p2)

i-,„ = MjJf(Ci*Zi,„, C'2*S2«i, - , C-p*ZpJ. We also can choose a subset of rules to perform inference based on a threshold value k, where X.e:[0, IJ, A rule R, can be fired if the degree of matching between the

379

observation facts and the antecedent portion of rule /?, is larger than or equal to X.. Otherwise, the rule R, can not be fired. For example, consider the following generalized mtxlus ptmens: Rule: IF X, is A, AND ... AND X, is A, THEN Y is fi Facts: Xi is A ' AND ... AND X/is A,* Consequence: Y is 5 «"

*

*•

Let A] , Aj ,..., Aj , and Aj be the vector representation of the fuzzy sets A], Aj,..., A/, and A) , respectively. Supposed X is a threshold value, XJE[0, 1 J, and the fuzzy set B has the following form: B ={(Vi, Zl), (V2, Z2),..., (Vm, Zm)},

where Zi e[0, 1] and 1 ^ / s /«. Let C = M ' « ( 5 ( A ' , AI ),..., S(Ai, A,)). If ca A, then the rule can be fired, and the deduced consequence of the rule is "Y is B"\ where B* ={(V,, C*Zi), (V2, C*Z2), ..., (V,„, C*2,^}.

If r < ^ then the rule cannot be fired. In this case, we let B*={(vi.0),(v2,0),...,(v,„,0)}. Consider the SISO fuzzy reasoning scheme shown in (1) or the MISO fuzzy reasoning scheme shown in (4). If there are some rules that were fired based on the threshold value A,, then the deduced consequence of these rules would be combined to get a deduced consequence "Y is HQ" of the fuzzy reasoning scheme. Finally, we can find an approximate linguistic term for BQ. This can be accomplished by using the linguistic approximation methcxl described in [11, p. 56] by matching BQ against a "linguistic term base", where all allowable linguistic terms are defined by membership function curves and the linguistic term which has the minimum Euclidean distance from BQ would be selected as the approximate linguistic term for BQ, where the Euclidean distance between any two fuzzy sets is calculated as follows. Let F and G be any two fuzzy sets of the universe of discourse U, where U = { «], M,, ..., «„} F = {(«!, /i), («2, A), ..., («„, /„)} G=

{(III, 5 , ) , (Uj,

& ) , ..., (M^

g„)}

and let F and G be the vector representation of the fuzzy sets F and G, respectively, where

380 F-

(/,. h

/«>

^ = fel. i'2. ••• .^„)The Euclidean distance between the fuzzy sets F and G is defined by d(F, G) = V ( / i - ^ i ) ^ + ( / 2 - & ) ' + ••• + ( / „ - & , ) ^ -

3. A Simple Example In this section, we use an example to illustrate the fuzzy reasoning process. Consider the following single-input-singic-output (SISO) fuzzy reasoning scheme: Rule l : I F X i s A i T H E N Y i s B i Rule 2: IF X is Ao THEN Y is B2 Rule 3: IF X is AT THEN Y is S, Rule 4: IF X is Aj THEN Y is B4 Rule 5: IF X is A5 THEN Y is B5 Fact: X is AQ Consequence: Y is £() where X and Y are linguistic variables, AQ, A [ , AT, A3, A4, and A5 are fuzzy sets of the universe of discourse t/, U={u\, «2,..., H14}, B\, Bi, BT„ B4, and B^ arc fuzzy sets of the universe of discourse V, V={i'i, vo,.... V14}. Assume that Ao=A3. These fuzzy sets are shown as follows: Ao= {(«,, 0), (H2, 0), (K3, 0), («4, 0), («5, 0), (H6, 0), (H7, 0), (H8, 0.5), (My, I), ( « K ) , 0.5), (HH.0),(«I2,0),(«I3,0),(H,4,0)}

A, = {{III, 1), {112, 1). («3, 1), (114, 0.5), (H5, 0), (Hf, 0), («7, 0), (MR, 0.5), («9, 1), («I0, 0.5), («ll,0),(«i2,0),(H,3,0),(Mi4,0)} A',= {(M,, {)), (Ml, 0), («3, 0), (H4, 0.33), (M5, 0.67), («6, 1), ("7. 0.67), (Mg, 0.33), (M9, 0), («io, 0), (M,|,0),(M,2,0),(M,3,0),(M,4,0)} A3 = {(M,, 0), (M2, 0), (M3, 0), («4, 0), {lis, 0), (M6, 0), (M7, 0), (Ms, 0.5), (Mg, 1), (MK), 0.5), ("11,0),(«12,0),(M13,0),(«14,0)} A4= {(«!. 0), (H2, 0), (H3, 0), (M4, 0), {II5, 0), (M6, 0), (M7, 0), (MR, 0), (Mg, 0), (HJO, 0.5), (Ma,l),(Mi2,0.5),(M,3,0),(Mi4,0)}

381 A5= {(«!, 0), (112, 0), («3, 0), («4, 0), (//5, 0), (H6, 0), (H7, 0), (lis, 0), (M9, 0), (Mio, 0), ( H „ , 0 ) , ( / / i 2 , 0 . 5 ) , ( / / n , 1),(«14, 1)} Bl = {(Vi, 1). (V2, I ) , (V3, 0.5), (V4, 0), (V5, 0), (V6,0),(V7,0),(V8,0),(V9,0),(Vio,0), ( V u , 0 ) , ( v , 2 , 0 ) , (V,3,0),(V14,0)} flT= {(vj, 0), (vo, 0), (V3, 0.5), (V4, 1), (vs, 0.5), (V6, 0), (V7, 0), (Vs, 0), (Vg, 0), (Vio, 0), (V„,0),(V,2,0),(V,3,0),(V,4,0)} ^3=. {(V), 0), (V,, 0),(V3, 0),(V4, 0.1), (V5, 0.4), (Vfi, 0.7), (V7, 1), (V8, 0.7), (Vg, 0.4), (v,o, 0.1), (v„,0),(Vi2,0),(Vi3,0),(Vi4.0)} fi4= {{Vl, 0), (V2, 0), (V3, 0 ) , (V4, 0), (V5, 0), (V6, 0), (V^, 0.1), (Vg, 0.4), (Vg, 0.7), (V,o, 1), (Vll,0.7),(V,2,0.4),(Vi3,0.1),(Vi4,0)} B5= {(V,, 0), (V2, 0), (V3, 0), (V4, 0), (V5, 0), (V(, 0), (V7, 0), (Vg, 0), (Vg, 0), (Vio, 0), (Vii,0,5),(Vi2, l),(Vi3, 1),(V,4, 1)}.

By using the vector representation method, the fuzzy sets AQ, A^ , A2, Aj,, A4, A5, Bi, B2, Bj,B4, and B5 can be represented by the vectors AQ,Ai,A2,Ai,A4,A^,B^,B2,B^, B^ , and B5 , respectively, where ^ = (0, 0, 0, 0, 0, 0, 0. 0.5, 1, 0.5, 0, 0, 0, O) AJ = ( 1 , 1 , 1 , 0 . 5 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ) ^ = (0, 0, 0, 0.33, 0.67, 1, 0.67, 0.33, 0, 0, 0, 0, 0, O) ^ = (0, 0, 0, 0, 0, 0, 0, 0.5, 1, 0.5, 0, 0, 0, 0> A^ = (0, 0, 0,0, 0,0, 0,0,0,0.5, 1,0.5, 0, O) ;^ = (0,0,0,0,0,0,0,0,0,0,0,0.5,1,1) BJ = (1, 1, 0.5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0> ^ = (0,0, 0.5, 1, 0.5, 0, 0, 0, 0, 0, 0, 0, 0, O) £^ = (0, 0, 0, 0.1, 0.4, 0.7, 1, 0.7, 0.4, 0.1, 0, 0, 0, 0> S^ = (0, 0, 0, 0, 0, 0, 0.1, 0.4, 0.7, 1, 0.7, 0.4, 0.1, 0> ^ = ( 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 . 5 , 1, 1, 1). The membership (unction curves are shown in Fig. 2.

382 memDership functions n

Ui

Al

U2

U:

U4

Us

Ue

U?

Ua

U9

UIOUP

U'2UI3U:

Va

V9

Vio V i : V : j

membership fupctions i.Bl

V;

V2

B2

V3

V4

B3

V5

V6

V7

V.3

Fig. 2. Membership function curves.

Based on the matching function S, we get

.V(^, Aj) =

S(A^, A;) = ,^

_ ^ 1 ^

=0

_'^"1^

^ 0.078

Mf/.r(A()« A(), A2 • A2)

Mi;7.r( Ao • Ag, A3 • A3)

VM

383

S(A^, A;)

-^"-"^

- 0.167

S(A^,^).

_'^L^^

= 0.

When the threshold value X, is set to 0, then all the rules can be fired. According to the previous discussion, we get the following results:

B\ = { ( V , , 0), (V2, 0), (V,, 0), (V4, 0), (V5, 0),

(V6, 0), (V7, 0), (Vg, 0), (V9, 0), (Vio, 0), ( v n , 0 ) , (Vi2,0),(v,3,0),(vi4,0)} B, = {(vi, 0), (V2, 0), (V3, 0.039), (V4, 0.078), (vs, 0.039), (V6, 0), (V7, 0), (Vg, 0), (V9, 0), (Vio, 0), (Vll,0),(Vi2,0),(Vi3,0),(Vi4,0)} R^ = {(vi, 0), (V2, 0), (V,, 0), (V4.0.1), (V5, 0.5), (V6, 0.8), (V7, 1), (Vg, 0.8), (V9, 0.5), (vio, 0.1), (Vll,0), (V|2,0),(Vi3,0),(Vi4,0)} li ={(Vl, 0), (V2, 0), (V3, 0), (V4, 0), (Vs, 0), (V6, 0), (V7, 0.0167), (Vg, 0.0835), (vg, 0.1336), (v,o, 0.167), (vii, 0.1336), (V]2, 0.0835), ( V B , 0.0167), ( v ^ , 0)} BI = { ( V I , 0), (V2, 0), (V3, 0), (V4, 0), (V5, 0),

(V6,0),(V7,0),(V8,0),(V9,0),(Vio,0), (Vll,0),(V,2,0),(Vi3,0),(Vi4,0)}. where /^, is the reasoning result from rule 7?,, I s / s 5 . Therefore, the deduced consequence of the fuzzy reasoning scheme is "Y is BQ", where ^ = B,' U ft U £^' U ^4 U B^ = {(vi, 0)" (V7, 0), (V3, 0.039), (V4, 0.1), (V5, 0.5), (V6, 0.8), (V7, 1), (i^, 0.8), (V9, 0.5), (vio, 0.167), (vn, 0.1336), ( v p , 0.0835), ( v ^ , 0.0167), (vi4, 0)}. The reasoning result of the SISO fuzzy reasoning scheme is represented by the dotted line shown in Fig. 3. Based on the linguistic approximation method, vvc can sec that the approximate linguistic term of the reasoning result/^ is B3.

384 membersMip f u n c t i o n s II

A1

Ui

U2

A2

UJ

U 4 U5

Ue

U7

Uo

U9

U:oUr

U)2Ui3Ui

membership fLnc'.ions iiBl

B2

Fig. 3. Reasoning result with threshold value set to 0.

When the threshold value k is set to 0.15, then by the previous discussion, only the rules Ry, and lix can be fired. In this case, we get B'I ={(vi, 0), (1-2, 0), (u,, 0), (V4,0), (V5, 0), (V(„0),(V7,0),(V8,0),(V9,0),(V,0,0), (Vll,0), (V'io,0),(v,3,0),(Vi4,0)} ft = {(V,, 0), (v., 0), (v^, 0), (V4, 0), (vs, 0), (V(„0),(v7,0),(v8,0),(v9,()),(vio,0), (V,],0), (V,2,0),(V,3,0),(Vi4,0)} /^' = {(vi, 0), (V2, 0), (v^, 0), (V4, 0.1), (V5, 0.5), (V5, 0.8), (V7, 1), (vs, 0.8), (V9, 0.5), (vio, 0.1), (Vll,0), (V,2,0),(Vi3,O),(Vi4,O)} li = {(vi, 0), (V2, 0), (v,, 0), (V4, 0), (V5, 0), (V(^ 0), (V7, 0.0167), (V8, 0.0835), (v^, 0.1336), (vjo, 0.167), (vii, 0.1336), (vp, 0.0835), (vn, 0.0167), (V14, 0)}

385 li, = {(V|, 0), (V2, 0), (\'3, 0), (V4, 0), (V5, 0), (V6, 0), (v^, 0), (vg, 0), (V9, 0), (vio, 0), (Vll.O), (V,2,0), (Vi3,0),(V]4,0)}. where B ' is the reasoning result from rule /?,, I s / s 5 . Therefore, the deduced consequence of the fuzzy reasoning scheme is "Y is BQ", where £b = 5i' U A U ^ ' U ^4 U /3^

= {(Vl, 0), (V2, 0), (V3, 0), (V4, 0.1), (V5, 0.5), (V6, 0.8), (V7, 1), (Vg, 0.8), (V9, 0.5), (v,o, 0.167), (vii, 0.1336), (vi2, 0.0835), (v^, 0.0167), (v^, 0)}. The reasoning result of the SISO fuzzy reasoning scheme is represented by the dotted line shown in Fig. 4. Based on the linguistic approximation method, we can see that the approximate linguistic term of the reasoning result BQ is Dsmembership functions

membership functions MBI

Vi

B2

V2 V3 V4

Vs

Vs

V7

Ve V9 V;o V i i V 1 2 V 1 1 V:.>

Fig. 4. Reasoning result with threshold value set to 0.15.

386

4. Conclusions Wc have presented a methodology to deal with fuzzy reasoning based on the matching function S. The single-mput-single-output (SISO) fuzzy reasoning scheme and the multi-input-single-output (MISO) fuzzy reasoning scheme were discussed in details. The proposed fuzzy reasoning methodology is conceptually clearer than the compositional rule ol inference approach. References I i I Z. Cao, A. Kandel, and L. Li, "A new mtxiel of fuzzv reasoning," Fuzzy Sets and Systems, \ ol. 36, pp. 311 -325, 199(). |2) T. C. Chang, K. Hasegawa, and C. W. Ibbs, "The effects of membership function on fuzzy reasoning," Fuzzy Sets and Systems, vol, 44, pp. 169-186, 1991. | 3 | S. M. Chen, "A new approach to handling fuzzy decisionmaking problems," IEEE Transactions on Systems, Man, and Cybernetics, vol. 18, no. 6, pp. 1012-1016, 1988. (4| S. M. Chen, J. S. Ke, and J. F. Chang, "Knowledge representation using fuzzy Petri nets," IEEE Transactions on Knowledge and Data Engineering, vol. 2, no. 3, pp. 311-319, 1990. [5] Y. Ezawa and A. Kandel, "Robust fuzzy inference," International Journal of Intelligent Systems, vol. 6, pp. 185-197, 1991. | 6 | J. Giarralano and G. Riley, Expert Systems: Principles and Programming, Boston: PWS-KENT Publishmg Company, 1989. |7| J. S. Kc and G. T. Her, "A fuzzy information retrieval system model," Proceedings of 1983 National Computer Symposium, Taiwan, 1983, pp. 147-155. [8j A. Kandel, Fuzzy Mathematical Techniques with Applications. Addision-Wesley Publishmg Company, 1986. (9| C. C. Lcc, "Fuzzy logic in control systems: fuzzy logic controller-part I," IEEE Transactions on Systems, Man, and Cybernetics, vol. 20, no. 2, pp. 404-418, 1990. 110] M. Mizumoto and H. J. Zimmermann, "Comparison of fuzzv reasoning methods," Fuzzy Sets and Systems, vol. 8, pp. 253-283, 1982. [111 K. J. Schmucker, Fuzzy Sets, Natural luinguage Compulations, and Risk Analysis, Rtx;kville, MD: Computer Science Press, 1984. |12| I. B. Turksen and Z. Zhong, "An approximate reasoning approach based on similarity measures," IEEE Transactions on Systems, Man, and Cybernetics, vol. 18, no. 6, pp. 1049-1056, 1988. (131 L- A. Zadch, "Fuzzy sets," Information and Control, vo]. 8, pp. 328-353, 1965. [14| L. A. Zadeh, "Outline of a new approach to the analysis of complex systems and decision process," IEEE Transactions on Systems, Man, and Cybernetics, vol. 3, no. 1, pp. 28-44, 1973. [15| L. A. Zadch, "The concepts of a linguistic variable and its application to approximate reasoning (1)," Information Science, vol. 8, pp. 199-249, 1975. |16| R. Zwick, E. Carlstein, and D. V. Budescu, "Measures of similarity among fuzzy concepts: a comparative analysis," International Journal of Approximate Reasoning, vol. l,pp. 221-242, 1987.

C H A P T E R 15

COMPUTING THE MULTIVALUED SHAPE OF A PATTERN CLASS Deba Prasad Mandal and C.A. Murthy Electronics & Communication Science Unit Indian Statistical Institute 203 B.T. Road Calcutta -700 035, India

I.

Introduction

The present chapter deals with the problem of determining the p a t t e r n class and its multivalued s h a p e / b o u n d a r y from sampled points (training samples). Once these are computed, some salient features of the class can then be extracted which are useful in making decisions about a course of action {e.g., identification, classification and p a t t e r n description) to be taken later. This will also reduce the storage requirement of the complete pattern class. It may be noted t h a t in most of the real life pattern recognition problems, the complete description of a pattern class is not known. Instead, a few sampled points are usually available which are assumed to represent the class. Hence determining the p a t t e r n class and its shape from sampled points is an important problem in p a t t e r n recognition. There are various approaches described in the literature for determining the shape of a p a t t e r n class from sampled points [1-6]. These methods are mostly heuristic in n a t u r e and they provide an exact boundary or shape of the p a t t e r n class. One of the inherent observations about these algorithms is that the b o u n d a r y of the class is restricted by the sampled points. This need not be true because the resulting boundary leaves certain regions not confined in it, although it should be. So, it is necessary to extend the boundaries to some extent to handle the possible uncovered portions by the sampled points. T h e extended portions should have the following two properties: (i) As the number of sampled points increases, the extended portions should decrease. (ii) T h e extended portions should have less possibility to be in the p a t t e r n class than the portions explicitly highlighted by t h e sampled points. T h e second property leads to define a multivalued or fuzzy (with continuum grade of belongingness) boundary of a p a t t e r n class. T h e basic concept of one of the existing methods [1-6] is described below in short for illustration. Edelsbrunner et al. fl] introduced the notion of the a-shape of a finite set of points, for arbitrary real a. This notion is a generalization of the convex hull. Given a set 5 of t points in a plane, the convex hull of S may be defined as the intersection of all closed half planes that contain all points of S. T h e a-hull of S is the complement of the union of all open discs of radius not less than 1 / Q (for arbitrary negative real 387 S.G. Tzafestas and A.N. Venetsanopoulos (eds.) Fuzzy Reasoning in Information, Decision and Control Systems, 387^18. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.

388

a) which contains no point of S. The shape of the set is determined using a-huUs. If the selection of a is proper then the shape of the set obtained by the method resembles the intuitive shape of the planar set. These Q-hulls have been found to be extremely useful in computational geometry [7]. The selection of a is very important for the method, but the authors did not provide any criterion for the selection of a. Also this method does not provide the multivalued shape of a pattern class. We have formulated here a methodology for providing multivalued (fuzzyl shape of a pattern class from a set of sampled points. The procedure can be viewea in two phases. Phase I is concerned with the decomposition of sample set into groups of nearly N-dimensional parallelepiped (henceforth parallelepiped) shape. The decomposition is based on the boundary variations (geometric complexity) of the pattern class found in the sampled points. A window approach has been incorporated here to find the boundary variations of the sample groups or sets. Phase II determines each of the subclasses corresponding to the groups separately, puts them together, and finds the multivalued shape of a pattern class. The effectiveness of the procedure has been demonstrated on some artificially generated data sets in IR^ and ff^. The practical applicability of the method has been demonstrated on a real life speech data set. The convergence of the estimated classes to the original ones has been verified using HausdorfF metric and a newly defined metric. In section / / , some basic concepts along with the block diagram of the proposed method are discussed. Section III deals with the procedure for decomposing a training sample set. Section IV provides an approach to determine the multivalued boundary of a pattern class. Experimental results are provided in section V. It is to be mentioned here that the sections / / / , IV and V concern with only two dimensional feature space. The extension of the procedure to higher dimensional space (M ) and its implementation to the pattern classes in ]F^ are provided in section VI. The convergence property of the proposed algorithm is discussed in Section VII. Section VIII deals with the conclusions. ^

II.

Some Basic Concepts and Block Diagram

Some of the basic concepts which are useful in developing the proposed method are initially stated here. The block diagram of the procedure is then provided.

A. a.

Some basic concepts Pattern class

In most of the real life problems, pattern classes are bounded. Thus the pattern classes considered here are all bounded. A formal definition of pattern class in M'^ is given below using topological and measure theoretic concepts. Definition 1 ; A set A C M'^ is said to be a pattern class [6] if (t) A is path connected compact, (u) cl{Int{A)) (Hi) Int{A)

= A, [cl means closure, Int means

interior]

is path connected and

[iv) X{6A) = 0 where 6A = An cl{A'^) and X is the Lehesgue measure on IR^.

389

The relevance of the properties (i), {ii), (Hi) and (iv) of Definition 1 is provided in [6]. Let ]B= {A : A satisfies Definition 1}. iB is the collection of all classes in Hi'^. Any A £ E is referred to as the pattern class. • b.

Accuracy factor

It has been argued in the previous section that the boundary of a pattern class obtained from its sampled points should be extended to some extent to highhght the possible uncovered portions of the class by the sampled points. An accuracy factor (St) based on the number of sampled points (<) is considered here for the said extension to manage the uncertainty. St satisfies [8] 1

.

1

,,.

juN < ^'< ^iuum

^^>

so that as t increases, i5( —> 0 and tS^ -+ oo. Since St decreases with the increase of t, the accuracy of the obtained boundary also increases with the increase of t. The selection of an accuracy factor is guidea by the inequality (1) and its justification can be found in [9, 10]. • c.

Coverage factors

Each individual sampled point represents a covered area of the pattern class in the feature space. Hence, in order to find the possible uncovered portions of the class by the sampled points, the boundaries are extended to some extent. The extended portions should have less possibility to be in the pattern class than the portions explicitly highlighted by the sampled points. To decide on the amount of extension, a factor, named as coverage factor, is defined below corresponding to each feature axis. Let Xi.,X2,- • • ,Xi,... ,Xt be the training samples where X,- = (x^j, a ; , j , . . . , x,-^,.. .,Xi^y and X, denotes the jih feature value of the fth sample. Let MAXj and MINj denote the maximum and minimum feature values respectively in the sample set corresponding to the jth {j = 1,2,..., iV) feature i.e., MAXj

— max Ix; 1 •'

1=1,2 t

and MINj

'

= ^

min | x , ) i=l,2,...,(

••

The coverage factor for the set of sampled points corresponding to the j t h feature, denoted by £j (j = 1,2,..., A'^), is defined as £j = {MAXj - MINj)

X St

(2)

where St is the accuracy factor. When the number (f) of sampled points increases, the value of the accuracy factor {St) decreases, and correspondingly the values of the coverage factors (e/s) also decrease and the accuracy of the boundary increases. • d.

Hole in a pattern class

A path connected and compact set is referred to here as a pattern class. If it happens that within the range of the pattern class or set, some portions do not belong to the

390

class, then the portions are referred to as the holes. The intuitive idea behind holes of a pattern class can be put mathematically by the following definition. Definition 2 : A -pattern class A is said to have k holes if A" = 5 U Ci U C2 U . . . U Cfc {%) B and Ci are path connected sets for i =

such that l,2,...,k,

(a) B is unbounded and Ci,C2,.. • iCk are bounded, {Hi) 5 U Cij U C,j U . . . U Ci^ is a disconnected set for 1 < t'l < J2 < • • • < Jr < ^, where 1 < r < k and {iv) di U Cij U . . . U Cir is a disconnected set I < ii < {2 < ... < ir < k, where 2
B.

Block diagram

The block diagram of the proposed multivalued shape determining procedure is shown in Fig. 1. It consists of two parts, namely the decomposition and the fuzzy processor. The decomposition section deals with the decomposition of the sample set into some groups of nearly parallelepiped shape. The fuzzy processor determines each of the subclasses corresponding to the sample groups separately and all these subclasses are then combined to compute the multivalued shape of the pattern class.

DECOMPOSITION Boundary Variation Calculator

Pattern Class Sub-divider

_L Membership Function Estimator

Boundary J Decider

H

,'^ Multivalued \ Pattern

V

Class

,/

FUZZY PROCESSOR

Figure 1 : Block diagram. The decomposition section consists of three blocks jis shown in Fig. 1. The hole detector block decomposes the training sample set with holes into groups to find the hole information. The boundary variation calculator block finds the values of the boundary variation factors corresponding to all possible boundary directions. These boundary variation values are analyzed in the pattern class sub-divider block to decompose (if necessary) the sample set into groups.

391

The theory of fuzzy sets [11-13] has been used in the fuzzy processor section to extend the boundary of the sample set and also to relate every point in the whole feature space to its possibility to be in the pattern class. The membership function estimator block decides about the compatibiUty/membership functions to represent each of the subclasses corresponding to the sample groups. The boundary decider block determines each of the subclasses separately, puts them together and finds the multivalued shape of a pattern class. Jk The following two sections (i.e., sections III and IV) describe in detail the operations of different blocks of Fig. 1 for pattern classes in IR^. A

III.

Decomposition

The decomposition section consists of three blocks, namely hole detector, boundary variation calculator and pattern class sub-divider. It takes the training samples of a pattern class as an input and decomposes the training sample set into groups if the sample set is found to be not nearly rectangular in shape. To obtain this decomposition, a procedure based on some overlapping windows is adopted here. Before describing the operations of various blocks of decomposition, the approach to generate the windows is furnished below. Generation of windows : The two features under consideration are referred to as the first (Fi) and second {F2) feature axes respectively. In the proposed approach, one of the axes is considered as the base axis and the other axis is considered as the height axis. Consequently the coverage factors cori'esponding to the base and height features are referred to as the base coverage factor (£{,) ana height threshold factor (eh) respectively. The sampled points are first of all arranged in ascending order according to the base feature values. The first window starts with the first sample of the ordered sample set and it includes all those samples one after another in ascending order until its base coverage length exceeds £),• Assume that the generated first window ends with the A;th sample of the ordered sample set. Then the second window will end with the (^-f l)th sample and to find the starting point of this window, it proceeds backward from kth sample until its base coverage length exceeds £;,. Similarly other windows are constructed by including one new sample at the end and excluding some samples from the beginning of the previous window such that the base coverage length would atleast be £(,. The last window ends with the last ordered sample i.e., sample with the highest base value. Thus, some overlapping windows of the sample points are generated utilizing the sample base values and the base coverage factor (ej). The maximum and the minimum height values are found from each of the windows and these are taken to be the upper (u) and lower (/) boundary values respectively of that window. The combination of tne upper boundary values highlights the upper boundary of the training sample set and the combination of the lower boundary values provides the lower boundary of the training sample set. The way in which the boundary of a training sample set is obtained using the aforementioned procedure for generating windows, is explained below considermg a typical pattern class [Fig. 2(a)]. A hypothetical training sample set is shown in Fig. 2(a), where the locations of samples in the feature space are shown by cross ( x ) marks. Initially, Fj is considered as the base and few windows are generated. In such windows, Fj is considered as the height feature. A typical window is shown by dotted lines in Fig. 2(6) where the samples in the window are shown by tick

392

{^/) marks. The boundary values corresponding to the lower and upper boundary directions (referred to as 2| and 2„ respectively) are also marked for the window. Based on these boundary values, the rough boundaries in the coded directions 2/ and 2„ are dra^vn in Fig. 2(c). To find the boundaries in the lower and upper directions of Ft (i.e., directions 1| and 1„), some windows are first of all generated considering F2 as the_base feature. A typical window with its sample points and boundary values is shown in Fig. 2{d). Pig. 2(e) shows the rough boundaries in the coded directions 1| and 1„.

s

)s

X

*

^

X X

^

a<

.t

*

"

m

Figure 2 : Illustrating tlie concept of boundary in M^. Combining the boundaries in figures 2(c) and 2(e), the complete boundary of a pattern class is obtained [Fig. 2(/)]. To incorporate the possible uncovered portions of the pattern class by the training set (Fig. 2(a)], these boundaries are extended to some extent (depending on the coverage factors). To visualize the said extension, the extended boundary of the pattern class is conceptually drawn in Fig. 2(/). The extended portion should have lower possibiUly to be in the class than the portions explicitly highlighted by the sample points. The extended regions decrease with the increase of the sample size (<). •

A.

Hole detector

The concepts about holes of a pattern class are provided in section II.A.d, The adopted procedure to find holes from a set of sampled points is discussed below. Procedure : The procedure considers Ft axis as the base feature and the F2 axis as the height. Correspondingly gj and £2 are considered as the base coverage factor (£i) and height threshold factor (e/i) respectively. Windows are now generated using the aforementioned approach. The sample points in each window are then arranged

393

in ascending order according to the height (Fj) sample values. If the difference of height values of any two consecutive samples within a window exceeds £/,, then a hole is assumed to be present between the said sample pair. Let h! and h" be the height feature values corresponding to two such sample points. To illustrate this finding, a pattern class with a hole is shown in Fig. 3 in which Px and Pj are the sample points which satisfy the above condition and the corresponding window is shown using dotted lines. To detect the hole, the sample set is decomposed into two groups according to whether the height values H.e., F2 values) are less than {h' + h")/2 or not. The decomposition leads to finding two groups where none of them possess a hole. In Fig. 3, the line with dashes indicates the split.

Figure 3 : A typical pattern class with a hole. The aforementioned routine is repeated until every sample group is found to be not containing any hole. When the subclasses corresponding to the sample groups are combined in the boundary decider block [section 4], the holes will be excluded from the final shape of the pattern class. It is to be realized that this procedure detects not only the holes elongated across F2 feature axis but also the holes elongated across Fi feature axis. Note that the proposed procedure can not detect the holes with sizes less than Si and/or £2 corresponding to the feature axes Fj and F^ respectively. Hence the detectable minimum hole size depends on coverage factors £1 and £21 which in turn depends on the accuracy factor {St) and finally depends on the sample size

it). It is also to be mentioned here that the procedure decomposes a training sample set not only for a hole, but also for some particular type of concave boundaries. For example, sample sets having shapes like 'C', ' ± ' or 'Z' may also be decomposed although these do not possess any hole. But this does not create any problem in the overall shape determining procedure, since such sample sets anyway had to be decomposed in the pattern class sub-divider block. A

B.

Boundary variation calculator

This block tries to detect the geometric structure of the pattern class from a set of sampled points. The boundaries in four perpendicular directions [Fig. 2(6)] are considered here to find the boundary variation values. Our approach to generate

394

the windows from a sample set and consequently to find the boundary values are described earlier. It may be recalled that the maximum and the minimum height values of each window are considered as the boundary values in the upper and lower boundary directions respectively corresponding to the height feature axis. The approach to calculate the boundary variations is discussed below. Calculation of boundary variations : To describe the approach, let us consider the boundary variation values in a particular direction, say d {d E {!/, !«, 2;, 2u}. The procedure to obtain the boundary values has already been discussed previously. Henceforth, it is assumed that there are w windows and their boundary values are //; {i = 1,2,... ,w). A boundary variation factor, denoted by V^ in direction d is defined as V, =

EW-'^.-i)' /el

(3)

where Ch is the height threshold factor for the direction d. Here the division factor E.\ is used to make the variation factor Vj unitless. Now let MAXH

and MINH

be the maximum and minimum of the boundary

values i/,'s (i = 1,2,... ,tu) respectively. If the difference of MAXH and MINH does not exceed £fi (i.e., if {MAXH ~ MINH) < eh] then the sample set is assumed to be nearly rectangular in shape. In such a case, the variation factor Vj is assumed to be zero i.e., make Vj = 0. Otherwise the sample set is considered as decomposable in the direction d. Initially, Fi axis is considered as the base to generate the windows and F2 axis is considered as the height to find the boundary variation factors V21 and V2 for the coded directions 2; and 2,^ respectively. Similarly, by reversing the roles of Fj and F2 axes above, the boundary variation factors Vi, and Vi„ for the coded directions 1( and lu are calculated. Thus, the boundary variation factors for the four boundary directions are obtained in this block.

C.

Pattern class sub-divider

This block analyzes the boundary variation factors to determine whether the training sample set is to be decomposed or not. If the sample set is to be decomposed then this block decomposes it into few groups. Before actually dividing the sample set, the decision about the direction of decomposition is to be made. To decide this, it finds the direction in which the value of the variation factor is maximum. That is, the direction D £ {1;, 1^, 2;, 2a) is found where VD = max{Vi„Vi„,V2„V2„}. If Vf) = 0, then the sample set is assumed to be nearly rectangular in shape and it is not further decomposable. Otherwise i.e., if VD > 0, then it is assumed that the sample set is not nearly rectangular in shape and it is to be decomposed into groups. Now from the direction of decomposition (i.e., D) the windows with their base and boundary values, and the corresponding height threshold factor Sh are recalled. The samples are then arranged in ascending order according to the base values. For making a cluster of windows, the maximum boundary value is found. The starting window for the cluster is taken as that window which have the maximum boundary value. The position of the starting window is noted. The following windows

395

from the starting windows are assigned one after another in the cluster until the differences between the boundary values of the current window and the starting window exceed the height threshold factor (ch). Similarly, the preceding windows are also put in the window cluster. The samples lying in the window cluster are assigned to the first sample group. The aforesaid routine is repeated on the remaining windows until all the windows are exhausted. This leads to the formation of window clusters. Every window cluster results in a group of sample points. Thus, the given training sample set is decomposed into a few group of sample points. The decomposition procedure is appUed on the sample groups repeatedly until all the groups are found to be nearly rectangular in shape. It is to be observed that the larger the number of groups, the greater will be the accuracy of the shape of the multivalued pattern class obtained. ^

IV.

Fuzzy Processor

The previous section dealt with the decomposition of a training set into few sample groups of nearly rectangular shape. The present section is devoted to obtaining the subclasses corresponding to these groups and combining them to get the estimated multivalued shape of the pattern class. This section consists of two blocks, namely the membership function estimator and the boundary decider. The concept of membership functions in the light of fuzzy set theory [11-13] is brought here to represent each of the subclasses corresponding to the groups separately. Membership functions have also been used to relate every point m the entire feature space to its possibility to be in the pattern class. The membership function estimator block finds the appropriate membership functions to represent each of the subclasses corresponding to the sample groups. The boundary decider block determines each of the subclasses separately, puts them together and obtains the multivalued shape of the pattern class.

A.

Membership function estimator

For any feature point, the possibility of being a member of a class is maximum if it lies in the centre of the class. As its distances from the the centre increases, the membership value decreases and ultimately goes to zero. Any function having this property may be considered as the representative membership function for the (^sub) pattern clciss corresponding to a sample group. As the TT function is well established to dictate this property [11-13], it is considered here as the representative membership function. Thus, the subclasses corresponding to the sample groups are characterized by different X functions across different axes of the form ir{x; ajt^, ^i^ , 0^.-, fit , Tuj, ) where k indicates the group number {k = 1,2,...,?7, r] denotes the number of groups); j indicates the axis number {j = 1,2); Q^^ is the peak value where the membership value is 1.0; /?/^ and /S^^, are the lower and upper most ambiguous points where the membership values are 0.5, and 7J^ and 7 ^ are the lower and upper end points beyond which the membership values are zero. The functional form

396 of such a TT function [10] is stated below: / a a \ \ S(x;^t,^ua) x ( a : ; a , ^ , , / J . , 7 , , 7 u ) = | il^^aj^,-,^) where S{x;a,b,c)

= <

0 1 (f5f) 1

if x < a \i x> a if X < a if a < X < 6

(4)

(5)

if X > c.

Note that unlike the standard S function [11-13], here 6 ^

a+c 2 •

, " ( « ( ^ , /St, ^ u , J<. ^

"4

"<=

4*

•%«.

Figure 4 : Pie (x) function. The structure of such a TC function is shown in Fig. 4. Given a particular group of samples and a particular feature, the domain of the TT function is [7;,7u]. It is assumed that the extended portions are [ft,Pi] and [/?u,7u], and the highlighted portion by the training sample set is [A»/3u]- These extended portions ultimately provide the multivalued shape of the pattern class. Determination of membership functions : To determine the membership functions (which are taken as n functions), the parameters of them corresponding to various sample groups are to be evaluated. Here each of the sample groups is considered separately. Let MAXkj and MINk^ be the maximum and minimum of the training sample set respectively corresponding to _;'th {j = 1,2) feature and A:th {k = 1,2,...,?/) sample group. Then the parameters of the TT function corresponding to j t h feature and A:th subclass (i.e., ^'th sample group) are assigned as follows: MAXk Qk, Ilk

+Mmk

=

MINU MINk

(6) 1.^,

= MAXk,+ej

;

j = l , 2 ; fc = l,2,...,Tj where Cj is the coverage factor for the j t h feature [Eq. (2)]. ^

397

B.

Boundary decider

In the previous subsection, membership functions corresponding to all the seimple groups along each feature axis are determined. Using these functions, each of the subclasses corresponding to the sample groups is estimated and those are finidly combined to obtain the estimated (multivalued) shape of the pattern class. All planar points in the feature space are labeled with their degree of possibilities to be in the cla.ss. To show the shape of a pattern class in the plane, the whole feature range is divided into small rectangles and these are referred to as the Feature Space Cells (henceforth FSCs). The size of all the FSCs are same and these are made as small as possible such that each FSC can be distinguished in the feature space. Thus, all these FSCs are labeled in terms of their possibility values to be in the pattern class. The FSCs with zero possibility vaJue are considered to be outside the class. The method of obtaining these possibility values is described below. Procedure: Let (xi,X2)' be a typical feature values of such a FSC. Suppose fXk. denote the membership value of the FSC corresponding to A:th [k = 1,2,...,?/) subclass (i.e., fcth sample group) and jth {j = 1,2) feature, fik^ is calculated from the corresponding TT function i.e., fik, = x [xf, ak,, Pi,^, P^,^, fi,^, 7uk^)

(7)

The combined membership of the FSC corresponding to A;th {k = 1,2,..., 7) subclass, say /Zfc, is defined as the geometric mean of /z^, and fik^ i-e., fik = {fik,Xfik,f^.

(8)

Now, the possibility, say 0, of the FSC to be in the estimated pattern class is defined as the maximum of the membership values of the subclasses. That is, 0=

max it=l,2,...,TJ

{ Uk ]

(9)

'^ ^^

^ '

'

Let r be the number of subclasses for which the combined membership values (/i/t's) of the said FSC are positive. To incorporate the effect of the neighboring subclasses with positive membership values in the estimated pattern class, tne value of 0 is increased to 6^^'^ for T > 1. That is, when the possibihty values of the FSC (/it's) are positive for two or more subclasses, then it indicates that the said FSC has the possibihty to He in those subclasses, which in turn increases the possibility of the FSC to be in the finally obtained pattern class. • A method has been described above to find the possibihty Vcdue {&) of a FSC to be in the pattern class. Note that 0 < 6 < 1. If the value of $ is zero, then the FSC is considered to lie outside the pattern class. Otherwise the FSC belongs to the pattern class with the possibility 6. To obtain the complete shape of the pattern class, the aforesaid routine is repeated for every FSC in the feature domain. Thus, all the FSCs are labeled with their possibility values to be in the pattern class, and as a result, the multivalued shape of the pattern class is obtained. 4

398

V.

Implementation and Results

In the previous two sections (i.e., sections III and IV), we have described the multivalued shape determining procedure for M^. The implementation and the usefulness of the procedure for M^ are demonstrated here on some artificiaUy generated data sets as well as on a real life speech data set.

A.

Artificially generated data

To show the effectiveness of the proposed system, different possible pattern classes were generated artificially and the procedure was implemented on them. The obtainea boundaries were found to be quite satisfactory in all the cases. Figures 5(a)-(d) show four typical pattern clstsses in M^. Note that the pattern cljiss in Fig. 5(c) is originally rectangular in shape, while others are not rectangular. The class shown in Fig. 5{d) has a hole whereas others do not have any hole. Training samples of size 50 are cnosen randomly from each of the four classes. The multivalued shape of these classes are shown in figures Q{a)-{d) with 0.15 as the accuracy factor (St). Initially in all the cases, the feature spaces are decomposed into smaJI rectsmgles (which are referred to as Feature Space Cells or FSCs) and the FSCs are labeled in terms of their possibility values {6) to be in the estimated pattern class. Recall that 0 < 6 < 1. The multivalued shapes are shown by various gray values in figures 6{a)-{d). Here the FSCs with gray value zero (absolute white) denote that the FSCs axe completely outside the pattern class. To obtain the crisp version of an estimated multivalued pattern class, the FSCs with possibility possibihty values (0) less than 0.5 may be considered to be outside the class. Observe that the algorithm has not decomposed the training sample set of the class in Fig. 5(c) because it is nearly rectangular in shape. But the sample set of the classes in figures 5(a) and 5(6) are decomposed into three and fourteen sample groups respectively. Note that the class shown in Fig. 5{d) has a hole. The algorithm initially decomposed the training sample set into two sample groups to detect the hole and finally the sample set has been decomposed into seven sample groups. The proposed technique is expldned below for the pattern class shown in Fig. 5(a). Fig. 7(ai) shows a training sample set corresponding to the class shown in Fig. 5(a). The algorithm first searches for the holes, but it could not find them. It then considered the boundaries in four perpendicular directions and found that the sample set can be decomposed in all the four directions. The boundary variation is found maximum in the coded direction 1/. Accordingly, the training sample set is decomposed into three sample groups with 12, 28 and 10 samples respectively [Fig. 7(03)]. Here the sample points of different groups are shown by three different characters. Each of these sample groups are then analyzed and it could not find enough boundary variation in any of the four directions for which these groups can be decomposed again. As a result, it obtained three groups [Fig. 7(a2)] of nearly rectangular shape. The three subclasses are determined based on the samples of these three groups. Finally these subclasses are combined to obtain the multivalued shape [Fig. 6(a)] of the pattern class in Fig. 5(a).

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B.

Speech data

To examine the practical applicability, the algorithm has been implemented on a set of Indian Telugu Vowel sounds in consonant-vowel-consonant context uttered by three speakers in the age group 30 to 35 years. Fig. 8(a) shows the typical feature space of six vowel classes /S/, / a / , / i / , / u / , / e / and / o / with 72, 89, 172, 151, 207 and 180 samples respectively corresponding to the features Fi and F2. Here Fi and F2 denote the first and second formant frequencies which were obtained through spectrum analysis of the speech data. Details of feature extraction procedure can be found in [13, 14]. The classes are seen to be overlapping and their boundaries cire ill-defined (fuzzy). The proposed algorithm has been applied on each of the six vowel classes separately where the total available data are assumed as the sampled points. That is, for the classes /6/, / a / , / i / , / u / , / e / and / o / , the number of sampled points are 72, 89, 172, 151, 207 and 180 respectively and correspondingly the accuracy factors are considered as 0.12, 0.12, 0.10, 0.10, 0.08 and 0.10 respectively. Fig. 8(6) shows the obtained multivalued shapes corresponding to the vowel classes / a / and / u / ; Fig. 8(c) shows the shape corresponding to the class / e / ; Fig. S{d) shows the estimated shapes corresponding to the classes / i / and /of, and Fig. 8(e) shows the shape corresponding to the class /6/. In figures 8(6)-(e), the FSCs with possibility values (6) > 0.5 are represented by the corresponding vowel characters; the FSCs with 6 satisfying 0 < ^ < 0.5 are represented by dots and the FSCs with ^ = 0 are shown as blanks. Observe that the estimated vowel classes are good representations of their original classes. 4

VI.

Extension to Higher Dimension

So far, we have dealt with the description and the implementation of the shape determining procedure for the pattern classes in JR . The extension concept of the procedure to the pattern classes in M'^ {N > 2) is furnished here. For the case oi JH}, we first decomposed the training set into some sample groups of nearly rectangular shapes and finally the subclasses corresponding to the sample groups are combined to obtain the multivalued shape of the pattern class (in JR^). The same spirit is adopted for ]R i.e., we first decompose (based on a window approach) the training sample set into some groups of nearly A^-dimensional parallelepiped shapes and these are combined to find the final shape of the pattern class (in IR^). The two parts of the shape determining procedure i.e.. Decomposition and Fuzzy processor are first of all described. The generalized decomposition concept is also elaborated for a pattern class in Iv. The algorithm is then demonstrated on some artificially generated pattern classes in ]R^.

A.

Decomposition

A pattern class in ]R is represented here by a set of sampled points. The decomposition section detects the geometric structure of the pattern class from the sample set. The iV features axes under consideration are referred to as the first (-Fi), second {F2), ..., A'^th {FN) axes respectively. Initially, the window generation procedure

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from a sample set in R'^ is described below. T h e proposed methodology for finding the shape is then described using the window generation procedure. Generation of windows : A window based approach is adopted here to find the boundary variations of a sample set in iR^ and the approach is more or less the same as t h a t described for M in section / / / . Initially, one of the feature axes is considered as the base feature and the corresponding coverage factor is referred to as the base coverage factor (ej). T h e training samples are first of all arranged in ascending order according to the base feature values. Then, depending on the base feature values and the base coverage factor Cb, t h e same procedure, as stated for M^, is followed here to generate windows. These windows are such t h a t base coverage length of each window would atleast be £t. Here for a particular window, the base feature values of the samples are assumed to be the same and t h e rest (A'-l) features for the samples may take any value. So the generated windows are visualized here to belong in ]R^~^ feature space; although the original sample set belongs to iR . I n this sense, the proposed method of generating windows always results in reducing the dimension of the sample set by one. In this context, note t h a t , the initial training sample set may itself be considered to belong to an TV-dimensional window. Based on the aforesaid concepts, the multivalued shape determining procedure is extended to JR^. T h e approach has been applied on a sample set in JR'^ in the following way. To start with, the feature Fi^ is considered as the base feature (axis) and correspondingly a few windows m M ~ are formed. For each of these windows, another axis F;, is considered as the base feature and windows in R'^~^ are formed. Repeating this process sequentially with Fi^, F;,, . . . , i^„_i as the base features axes, some one-dimensional (l-D) windows are generated eventually. T h e values of the left out feature F^^ are considered here as the height values. T h e maximum and the minimum height sample values are found for each window and these are taken to be the upper and the lower boundary values respectively for t h a t window. T h e combination of the upper boundary values of all l-D windows, generated by sequentially taking i ^ j , F^j, . . . , Fi^_^ as the base axes, highlights the upper boundary of the training sample set in the Fi^^ feature direction. Similarly, the combination of the lower boundary values provides the lower boundary of the sample set across F,-„ feature axis. a.

Hole

detector

T h e operation of this block is same as that described in section IILA. T h e procedure sequentially considers Fi, F2, . . . , F^-i as the base features (and correspondingly £1, £2, . . . , £iv-i as the bcise coverage factors) to generate few l-D windows with F^ as the height feature. Here the coverage factor eN across the axis F^ is referred to as the height threshold factor. T h e samples in each window are then arranged in ascending order according to the height ( F ^ ) sample values. If t h e difference of height values of any two consecutive samples within a window exceeds ejv, then a hole is assumed to be present between the said sample pair. Let h' and h" be the height feature values corresponding to two such sample points. To detect the hole, the sample set is decomposed into two groups according to whether the height values (i.e., FN values) are less than {h' + h")/2 or not. T h e above routine is repeated until every sample group is found to be not containing any hole. When the subclasses corresponding to the sample groups are combined,

405

the holes are excluded from the final shape of the pattern class. • b.

Boundary variation calculator

To find the boundary variations of the sample set in M , 2N perpendicular directions (coded as 1;, !„, 2(, 2^, . . . , A^;, N^) corresponding to the lower and upper boundary directions along the N feature axes are considered. The A^ features are denoted here by F „ , Fi^, . . . , Fi^ such that 4 £ { 1 , 2 , . . . , A'^} for k = 1,2,...,A'^ and 4 7^ 4 ' for k ^ k'. The procedure to find the boundary variation values is described below for a training sample set in IR . Though the notation used here may seem to be cumbersome, it may be noted that the procedure is the natural extension of the 2-D case (described in section III.B). Initially assuming the feature /ii as the base (and correspondingly e„ as the base coverage factor), some windows are generated from the total training set. Let the number of windows generated be denoted by 9,-,. As mentioned earlier, in each of these ^i, windows, the F,-, feature values of the samples are considered to be same and so the samples are visualized to belong in JR with Fi^, Fi^,..., Fj^ as the axes. Let a particular window, say jjth one, be denoted by W^^ where j'l — 1,2,... ,^ij. Now assuming the feature Fi^ as the base (and correspondingly £,j as the base coverage factor), the sample set of the window W^' (E Ft "*) results in qf^,^ •. windows in (A'^2) dimensional space with i^-j, Fi^, ..., F,-^ as the feature axes. Similarly, considering sequentially the features ^^3, Fi^, ..., Fi^_^ as the bases (and correspondingly e^j, £ii, • • •, £-iN-2 ^® ^^^ respective base coverage factors), some 2-D windows will be generated and the number of such windows is denoted by 9^^/( (/''j (, _ )) \\ (where j i = 1,2,..., qil(i^t}ik-AiJl--))' forfc= 1,2,..., A^-3). These windows are visualized here to belong to M^ with Fi^_^ and F^^ as the axes. Thus the original sample set (belonging to IR ), after the aforementioned operations, gives rise to a collection of 2-D windows. The feature Fijj_^ is now assumed to be the base (and correspondingly e,-^_, is assumed as the base coverage factor) so that the samples of the jV-2th window Wi'^fff^-^-i^ results in ^f/f/",'/''-','/"" 11 u I'D windows with F^ as the feature axis. For the samples belonging to these l-D windows, the values of Fi^, are now considered as the height values. The maximum and the minimum height sample values are found in each of the l-D windows and these are taken as the upper and the lower boundary values respectively for the respective windows. Let ^i^(h(...iiN-2)-)) denote the boundary value of the kth {k — 1,2,..., 'l^llhtiiN-fiiNW-.))') ^'^ window (generated by taking Fi^_, as the base) for the feature Fi^ in the direction d {d E {/, u} where / and u stand for lower and upper boundary directions respectively). For each of the 9 i \ / ( (; JV; _ ii )) 2-Z) windows, the upper and the lower boundary variation factors, denoted by V^'^,- / / •'^^ ,), are

406

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and MINH

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407

1-D windows ia F2 feature space. Finally the boundary variation factors of F2 in the lower and upper boundary directions corresponding to their 2-D windows are obtained. So for all the above gi 2-D windows, the boundary variation factors V^"^\ F,^3j«=^% V,%'--'\ n w " = " (ii = l , 2 , . . . , f t ) a r e c a l c u k t e d . If F2 is initially taken to be the base, the original sample set results in q2 2D windows with Fi and F3, as the feature axes. A typical of such windows is marked in Fig, 9(c). The 2-D view of the marked window is provided in Fig. 9(d). Corresponding to each of the qj 2-D windows, the boundary variation fac3 : I : i2 ^3 : « t h I •• h ^2(3)" ' ^^ (j2 = 1! 2 , . . . , <|2) are obtained. tors F,2(1) ''2(1) ! •-2(3) 1

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Figure i : Concept of windows for a pattern class in JH^. If F3 is first considered to be the base, the initial sample set results in qs 2-D windows «'ith Fi and F2 as the feature axes. A typical windows is marked in Fig. 9(e) and its 2-D view is provided in Fig. 9(/). Corresponding to each of the §3 2^ windows, the boundary variation factors K 5 A ' ' * , ''^li)" ' ^^ ^*(2)' ' ""^ •^3(2) (is = 1,2,..., ga) are determined. Thus, there are in total 4(gi + f2 + fs) boundary variation factors corresponding to a sample set in I^. All these variation factors are analyzed in the next block i.e., pattern class sub-divider block. • c.

Pattern class sub-divider

This block analyzes the boundary variation factors to determine whether the training samole set is to be decomoo.sed or not. To decide this, it finds the maximum of aJl

408

neaxly parallelepiped in shape and it is not further decomposable. Otherwise it is assumed that the sample set is not nearly parallelepiped in shape and it is to be decomposed into few groups. Corresponding to the 2-D window and the direction of the maximum variation factor, the l-D windows with their ba^e and boundary values, and the coverage factor corresponding to the height feature (referred to as height threshold factor e/i) are marked. Based on these values, the decomposition is made. The samples in tne particular 2-D window are then arranged in siscending order according to the base values. For making a cluster of l-D windows, the maximum boundary value is found. The starting window for the cluster is taken as the l-D window having the maximum boundary value. The position of the starting window is noted. The following 1D windows from the starting windows are assigned one after another in the cluster until the differences between the boundary values of the current l-D window and the starting window exceeds the height threshold factor (e^). Similarly the preceding 1D windows are also put in the window cluster. The maximum (say, MAXi,) and the minimum (say, MINi,) base values of the samples in the window cluster are found. Now from all the samples in the considered sample set/group, the samples with the base lying between MINb and MAXt are assigned to the first sample group. The aforesaid routine is repeated on the remaining l-D windows until all the marked l-D windows are exhausted. This leads to the formation of window clusters. Every window cluster results in a group of sample points. Thus, the given training sample set is decomposed into a few groups of sample points. The decomposition procedure is applied on each of the sample groups repeatedly until all the groups are found to be nearly parallelepiped in shape, ft

B.

Fuzzy processor

A training sample set is decomposed in the previous section into few groups of nearly parallelepiped shape. Here the subclasses corresponding to these groups are determined separately and finally these are combined to obtain the multivalued shape of the pattern class. a.

Membership function

estimator

The subclasses corresponding to the groups are characterized by different JT functions across different axes of the form 7r(x; ajt^, 0i^ , Pu,, , It/, , 7ut ) [Eq. (4) and Fig. 4] where k indicates the group number {k = 1,2^... ,r], -q denotes the number of groups); j indicates the axis number (j = 1,2,..., N). The meaning of the parameter of the ir functions are the same as mentioned in section IV.A in relation to the pattern classes in JR^. Using Eq. (6), the parameters are determined for each subclasses corresponding to sample groups and each feature axes. • b.

Boundary decider

To show the shape of a pattern class in the feature space, the entire feature range is divided into small units of parallelepiped shape and these small units are referred to as the Feature Space Cell or FSC. The size of all the FSCs are same and these are made as small as possible such that each FSC can be distinguished in the feature

409

Space. All these FSCs are labeled in terms of their possibility values {0} to be ia the pattern class. Let (a^i, a;2,..., x^)' he a typical feature value of such a FSC. The membership value (nkj) of the FSC corresponding to tth (k = 1,2,,..,»|) subclass (i.e., tth sample group) and j t h (j = 1,2,... ,iV) feature is calculated from the corresponding w function using Eq. (7). The combined membership (^i) of the FSC corresponding to i t h {k = l,2,...,i]} subclass is defined as the geometric mean of pk/s i.e.,

m=

{p-ki X |i|.3 X . . . X Ukuf^^-

The . . „ possibility r - " - — . - value ~ — (0) v / of the FSC is then determined using° -Eq. - (^)- Depending •alue of the FSC in the neighboring subclasses, the value of 0 on the membership values is similarly modified as done for M^. Thus, by labeling all the FSCs of the feature domain with their possibility values, the multivalued shape of a pattern class in JR' is obtained. The implementation of the extension concepts is provided below. •

Figure 10 : (a) A pattern class; (b)-(d) corresponding estimated versions with 6 > 0.5, 6 > 0.25, # > 0 respectively.

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C.

Implementation and results

The effectiveness of the proposed procedure has already been demonstrated in section V on some artificially generated data sets and also on the real life speech data which were in M^. The behaviour of the generalized procedure is demonstrated here with three artificially generated pattern classes in JR^. The classes are shown in figures 10(a), 11(a) and 12(a). For all 3-D classes, the isometric views are provided. Note that Fig. 12(a) shows only the lower bisecting (with respect to F^ feature) portion of a pattern class which has the same external view with the class in Fig. 10(a). Actually this class is having a hole and externally the hole can not be shown. For demonstrating the hole detecting capabilities of the proposed algorithm, only the lower bisecting portion of a pattern class is displayed in Fig. 12(a). Training samples of size 150 are chosen randomly from each of the three classes and correspondingly the accuracy factor {6t) is assumed to be 0.20. Note that the extracted classes are multivalued. Hence, in order to demonstrate the concept of the multivalued shapes, three levels of estimated classes based on the possibility values {$), namely 0 > 0.5, 0 > 0.25, 0>0 are only shown. Figures 10(6), 10(c) and 10(d) show the estimated shapes with 0 > 0.5, 0 > 0.25, 0 > 0 respectively corresponding to the pattern class in Fig. 10(a). Figures 11(6), 11(c) and 11(d) show the estimated shapes with 0 > 0.5, 0 > 0.25, ^ > 0 respectively corresponding to the pattern class in Fig. 11(a). Figures 12(6), 12(c) and 12(d) show the lower bisected portion of the estimated shapes with 0 > 0.5, 0 > 0.25, 0 > 0 respectively corresponding to the lower bisected portion as shown in Fig. 12(a) of a pattern class.

VII.

Convergence with Sample Size

Convergence of the estimated pattern class to the original pattern class is shown here experimentally and analytically for any dimension N >2. For any shape determining approach based on sampled points, the accuracy of the estimated shapes, in general, should improve with the increase of the size (number) of the sampled points. It will be shown in this section that the proposed multivalued shape determining procedure also has this property. As the sample size (i) increases, the value of the accuracy factor (St) decreases and accordingly the accuracy of the estimated multivalued shape increases.

A.

Experimental verification

Two artificially generated pattern classes have been considered in this section to demonstrate the convergence property of the proposed algorithm. The first one is a disc [Fig. 13(o)] with radius 2 and centre at (3,3). Five different sets of data are chosen randomly from it with sizes 50, 100, 150, 200 and 250. The algorithm was applied assuming these five data sets as sample points with accuracy factors (Si) 0.16, 0.13, 0.10, 0.08 and 0.07 respectively. The corresponding estimated multivalued classes are shown in figures 13(6)-(/). The other considered pattern class is a sphere [Fig. 14] with radius 2 and centre at (3,3,3). Four different sets of data are chosen randomly from it with sizes 150, 300, 500 £ind 1200 respectively and the values of 6t are considered as 0.20, 0.16, 0.13 and 0.10 respectively. Here also three levels of extracted classes are shown corresponding to 0 > 0.5, 0 > 0.25, 0 > 0. Figures I5{a)-{d) show the estimated classes with

412

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413

Figure 14 : A spherical class.

Figure 15 : (a)-(cl) Estimated versions of the class in Fig. 14 with if > 0.5 based on 150, 300, 500 and 1200 samples respectively.

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9 > 0.5 based on the selected sample sets of sizes 150, 300, 500 and 1200 respectively corresponding to the pattern class in Fig. 14. Figures 16^a)-( 0.25 based on 150, 300, 500 and 1200 traming samples respectively. Figures n{a)-{d) show the estimated classes with ^ > 0 based on 150, 300, 500 and 1200 training samples respectively. It can be seen from these results that as the sample size (i) increases the estimated classes are gradually converging to the original pattern class. A

B.

Criteria for goodness of fit

The aforementioned convergence property is also verified analytically using two distance measures (metrics). One of them is the HausdorfF metric and the other one is a new similarity metric {Sim) that has been defined here. It has been shown that the values of both the metrics tend toward zero as t ^ oo. a.

Hausdorff metric

Normally, to find the similarity between sets, a distance measure is often used. Hausdorff metric [15] has been used here for this purpose. Let {X, d) be a metric space. For any compact subset of ,4 of A^, define S{y,A)

=

ini d{x,y)

where inf means infimum. Note that 6(y, A) is finite and BXQ G A such that S(y, A) = d{xo,y). Now the definition of the Hausdorff metric is given below. Definition 3 [15]: Let A and B he two compact subsets in M^ . Then the distance between A and B, denoted by Dist[A,B), is defined as Dist{A, B) = max I sup S{x,B) I xeA

, sup 6{y,A) yeB

>

(11)

J

the sets A and B are non-empty. Here sup and max stand for supremum and maximum respectively. • If the original sets are assumed to be finite, the sup and inf can be replaced by max and mm (minimum) respectively and then Dist{A, B) is defined as Dist{A, B) - max \ max S(x,B) where

6{x,B)

=

min

d{x,y)

and

S(y,A)

=

min

d(x,y).

, max 6{y,A)

\

(12)

This distance measure Dist{A, B) is considered here as one of the criteria for goodness of fit, where A is considered as the boundary of the estimated set or class and B is considered as the boundary of the original class. This distance measure has initially been applied on the five estimated sets or classes [figures 13(6)-(/)] with the original set [Fig. 13(a)] in the following way.

416

The boundary of the disc [Fig. 13fa)] is approximated by 180 equally spaced points. This set of 180 points is considered here as the set B. Note that the estimated classes are multivalued. Hence, in order to apply this measure, three levels of estimated boundary based on the possibihty values (#), namely 0 > 0.5, (? > 0.25 and # > 0 are considered. The values of the Dist measure are shown by a graph in Fig, 18(a).

I i > O.S • > 0.25 jl > 0

> > 0.5 > .1 0-25 S > 0

HI --0 Sampl« Size (Estimated Class)

son

1200

Siie {Estimated CiMJ)

(a) Figure 18 : Values of Dist measure (a) between the estimated classes in figures 13(l»)-(/) and the actual class in Fig. 13(n); (b) between the actual class in Fig. 14 and its estimated versions. The distance measure Dist has also been applied on the estimated sets or classes [figures 15(a)-(d), 16(a)-(d) and 17(a)^(d)] with the original set in Fig. 14. ^ The boundary of the sphere is approximated by 5675 equally spaced points and this set of 5675 points is considered here as the set 5 . Here also three levels of estimated boundary based OE the possibihty values (fl), namely 0 > 0.5, 0 > 0.25 and 0 > 0 are considered. The values of the Dist measure are shom'n by a graph in Fig. 18(6). * b.

A netfl metric Sim

Note that the Hausdorff metric reflects the overall similarity between two closed sets. In order to incorporate the similarity of each of the elements of the sets, a new measure has been defined below. Definition 4 : Let A and B be two finite sets with t^ and Ig elemc»fa respectively. Then a similarity measure between A and B, denoted by Sim{A,B), is defined as Stm{A,B)

= j

^%,B) + j

X^%,^) + D«l(A0)

(13)

Here the first term denotes the average similarity of the dements of A to B, the second term denotes the average similarity of the elements of B to A and the last term Dist{A,B) denotes the overall similarity between A and B. Sim is a metric [9, 10]. •

417

The defined metric Sim has been considered as another criterion for goodness of fit in the proposed shape determiniDg procedure. It has initially been applied between each of the five estimated sets (with three levels of boundaries as # > 0.5, 0 > 0.25 and 0 > 0) [figures 13(ii)-(/)] and the original set [Fig. 13(a)] in the same way as the previous case. The values of the Sim measure are shown in Fig. 19(a).

9 > O-i S > 0.25 « > 0 i » > 0,5 9 > 0,25 # > 0

Pi

0

ISO

•••

S i m p t a Siee ( S j l i m a t e d Cl»s»)

(4

- ; 3

150

_.

._

.••..)

Samplea SsEif (Ealirnated Class)

W

Figure 1§: Values of Sim measure (a) between the estimated classes in igures 13(i»)-(/) and the actual class in Fig. 13(a); (b) between the actual class ifl Fig. 14 and its estimated versions. The metric Sim has finally been apphed between the pattern class in Fig. 14 and its estimated multivalued classes [figures 15(a)-(d), 16(a)-(d) and 17(a)-(d)]. The values of the Sim measure are provided in Fig. 19(6). Hence, the convergence property of the proposed shape determining procedure is established both experimentally and analytically. •

VIII,

Conclusions

In this article, we have described an approach to compute the multivalued shape of a pattern class from its sampled points. The concept of fuzzy set theory has been found to be appropriate here. In estimating the shape, the portions not covered by the sampled points are assigned some fuzzy membership values denoting the degrees of their belonging to the actual class. Therefore, unHke the conventional approaches the proposed method does not attempt to provide crisp boundary from incomplete sample set. The effectiveness of the methodology has been demonstrated on some artificially generated data sets (in JR^ and JR^) and also on the real life speech data. The multivalued shapes can be converted to the usual crisp versions by considering only the feature points (FSC) with possibihty value (0) > 0.5, say to be within the classes. The convergence of the estimated shape to the original one has been verified both experimentally and analytically. la this context, a new metric Sim has been defined which can be used to find the similarity between any two finite sets. The parameters (like accuracy factor, coverage factors etc.) required for determinme the shaoe of a class are determined automatically from the sampled points. One

418

represented as a collection of some nearly parallelepiped shaped subclasses. This concept has been found very useful in developing a multivalued recognition system

[16]. 4k

Acknowledgement T h e authors gratefully acknowledge Professor Sankar K. Paliov his valuable guidance and suggestions in this work.

References Edelsbrunner, D. G. Kirkpatick, and R. Seidel, "On the shape of a set of points in [1 aH.plane," IEEE Trans. Inform. Theory, vol. IT-29, pp. 551-559, 1983. [2 R. A. Jarvis, "Computing the shape hull of points in the plane," in Proc. IEEE Comp. Soc. Conf. on Patt. Recog. and Image Process., pp. 231-241, 1977. [3] S. K. Akl and G. T. Toussaint, "Efficient convex hull algorithm for pattern recognition applications," in Proc. 4th Int. jt. Conf. on Patt. Recog., Kyoto, pp. 483-487, 1978. [4] J. Fairfield, "Contoured shape generation forms that people see in dot patterns," in Proc. IEEE Conf on Cybem. and Soc., pp. 60-64, 1979.

[s; G. T. Tousscdnt, "Pattern recognition and geometrical complexity," in Proc. 5th Int. Conf Patt. Recog., Miami Beach, Florida, pp. 1324-1347, 1980.

[6; C. A. Murthy, On consistent Estimation of classes in M^ in the context of cluster analysis. PhD thesis, Indian Statistical Institute, Calcutta, India, 1988. [7: F. P. Preparata and M. I. Shamos, Computational York: Springer Verlag, 1985.

Geometry : An Introduction. New

[8] U. Grenander, Abstract Inference. New York: John Wiley, 1981. [9] D. P. Mandal, A Multivalued Approach for Uncertainty Management in Pattern Recognition Problems Using Fuzzy Sets. PhD thesis, Indian Statistical Institute, Calcutta, India, 1992. [lo: D. P. Mandal, C. A. Murthy, and S. K. Pal, "Determining the shape of a pattern class from sampled points in iR^," Int. J. General Systems, vol. 20, pp. 307-339, 1992. [n

L. A. Zadeh, "Fuzzy sets," Information

& Control, vol. 8, no. 3, pp. 338-353, 1965.

L. A. Zadeh, "An outline of a new approach to the analysis of complex systems and [12; decision processes," IEEE Trans. Syst., Man & Cybem., vol. SMC-3, pp. 28-44, 1973. S. K. Pal and D. Dutta Majumder, Fuzzy Mathematical Approach to Pattern Recog[13: nition. New York: John Wiley (Halsted Press), 1986. [14 S. K. Pal and D. Dutta Majumder, "Fuzzy sets and decision making approaches in vowel and speaker recognition," IEEE Trans. Syst., Man & Cybem., vol. SMC-7, pp. 625-629, 1977. [is: K. Kuratowski, Topology, vol. I. New York: Academic Press, 1966. [16: D. P. MEUidaJ, C. A. Murthy, and S. K. Pal, "Formulation of a multivalued recognition system," IEEE Trans. System, Man & Cybem., vol. SMC-22, pp. 607-620, 1992.

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PART 5

APPLICATIONS

CHAPTER 16

ROBOTIC CONTROL USING FUZZY LOGIC AND PARALLEL PROCESSING M J . H e n d e r s o n and K.F. Gill Department of Mechanical Engineering The University of Leeds W o o d h o u s e Lane, Leeds LS2 9JT, England

1

Introduction

In the last twenty five years a number of new control techniques, such as fuzzy logic, have been introduced [1]. Fuzzy logic has met with some success in the process control industries and a number of papers have been published, typically Daley [2]. The work presented here enhances the published work by investigating the application of parstllel processing to real time fuzzy logic in a project carried out by the authors. INMOS transputers [3] are used as the building blocks for the multi-instruction, multi-data parallel (MIMD) computer employed in this project.

2

System Hardware

The project utilizes a small Mitsubishi industrial robot, the RM-501 [4], with the proprietary control unit removed to allow direct access to the actuators, optical encoders and joint boundary detection switches. Counter cards have been introduced to register the output from the optical encoders employed to measure joint position and each actuator is controlled directly from a transputer, with voltage as the manipulated variable. A three INMOS transputers network resident in a SENSION B004 system [5], was atdopted because it would allow adequate software developement for the project envisaged. The "operating system" was the Transputer Developement System [3] with Occam II [6] as the programming language. A 8048&'microproce88or based IBM compatable computer served as the front end to support the transputer network. The three transputers used in the network are a T400 transputer resident upon the GBUS board [6] providing a memory mapped interface to the robot, a T&OO root transputer and a T800 host transputer. The host transputer is connected by an interface card to the IBM computer used to provide the necessary user interface and disc storage facilities. The digital to analogue cards (DACs), counters and joint boundary detection switches are connected to Peripheral Interface adapter (PIA) cards which are memory mapped onto the INMOS T400 transputer by the GBUS card, allowing the PIA cards to be accessed by the appropriate read and write operations to specific locations in memory. The PIA cards conform to the interface protocol implemented in the GBUS and are called GESPIA cards. A schematic representation of part of the system used is shown in Figure 1.

3

Fuzzy Logic C o n t r o l

Zadeh [1] first introduced the concept of fuzzy conditional statements and using these statements Mamdani [7] developed his fuzzy logic controller. The Mamdani approach to the construction of a fuzzy logic controller has formed the basis of the approach adopted by the authors. In this work the domains of discourse over which the fuzzy logic controller is defined are integer and finite. The input domains are error ( UE ) and rate of change in error ( URCE ) with the output domain being change in control action ( Ucv )• The integer range was selected for practical reasons as the information read from the robot optical encoder counter cards and written to the DACs is 421

S.G. Tzafestas and A.N. Venetsanopoulos (eds.) Fuzzy Reasoning in Information, Decision and Control Systems, 421—449. © \99A Kluwer Academic Publishers. Printed in the Netherlands.

422 TOANSPUTEH COMMUNICATION LINKS

MEMORY MAPPED INTERFACE

SESPIA'CWO-

Figure 1

GESHA-CARO'^

The Interface

integer, If desired, the discttssioii can readily be geoerdized to domains delned over the reaj number universe. At each sample interral it there afe two controller inputs, error (e(i) e Mjs) and rate of change in amt (rce(t) 6 UHCE) with change in manipulated control variable (w(i) e Mcc?) « the output. Eate of change in erioi and not change in error is used in order to avoid the need to rcdeine the fuziy sets when a differeot sampling rate (1.) is used. The variables e{k) and rce(k) are calcmlated from the demand {d(k)} and the joint state (t(i)), ndng the expressions: e(lt) = cl(Ji:) - »(t) and rce(*) = - ^ ^

=

(1) '-

(2)

By adopting u ( i ) m the output there is no risk of large changes in the value of the manipnlated controlvariable, even in the presence of a "rogue" rule. The manipnlated control variable (c(i)) is evaluated from the expression: (3)

c(it) = J 3 " ( ' ) *''*••*' "W) = 0

A fuszy logic controller is built mp from a number of rules, typically in this project between 69 and 289 rules. The i'th rule has the form: IF ( 4i)

IS P O S I T I V E . B I G ) AND ( rce(li:) IS P O S I T l T V E . S M A t l . ) THEN ( c»(jt) IS KEGATIYE.MEDIUM )

where P O S I T I V E . B I G , P O S I T I V E . S M A I , ! , and N E G A T I V E - M E B I D M are Bnguistic labels for the fuzzy sets defined over the UB,UMCE and Ucu domahis of discouse. The output at sample interval i- is FCi{k) £ Ucu- This ii calcnlated from the membership grades of the input and output fuzzy sets which are labelled Ei,, RCE,^ and CUi^ . Adopting the convention that fuzzy set charactetiBtic functions are represented by fi(.} with the appropriate fuzzy set symbol written as a subscript, tlie characteristic function |iFC,(i:) which defines the membership grade of the output at each point en e Ucu is defined using the minimum operator /\: (4)

Let h,(k) = / \

[m,M^)lncE,Me{h])]

then l'FCiik){m) = f\[hi(k),itctj,^(cu)]

VcueUcu

(5) (6)

423

The output at sample interval k ftom all the rules are combined iiaiiig the maximiiin operator y to give a single fuiiy set, FGH) £ Ucu- The deterministic cbaoge in the output control variable u{k) is usually calculated from FC{k}. The membeirsbip grades of FC{k) are defined by the expiession: K i=\

The deterniinlstic change im control vaiiable output at sample interval t is calculated from FC{k) by the defrazification process, DF{,)t u,{k) = DF(FC(k)) (8)

S.l

Fuzzy Set Shapes

Various methods have been investigated in the literature to determine if real time operation can be achieved with a fuzzy logic contioller. The avenues of approach explored include requantised domains and look up tables [8] and simplified fujissy seta and rules |9], success has been achieved for moderate sample rates, far higher sample rates are requited in robotics. The amtliow hawe investigated a novel approach where by the characteristic function is assumed to be of a known form, termed a template.With this protocol it is only necessary to store the numbers which give the exact wale and location in a domain of the characteristic function template. By cmployinh a template representation, for the characteristic funation, "real time' caknlations becomes more realizable. The template protocol may be used to store information describing many shapes. Unless otherwise stated, the protocol of an ordered quadruple describes a trapasoid. E x a m p l e 1 Consider Ikt example of the cftararferisfie fimctian shown in Figure S, defined ever an integer iemsim of discourse. The pammeiers required to define this faneiion asssming a tmpazoiial temflaU are (5,14,25,37).

MEMBEHSHIP GRADE

DOMAIN OF DISCOURSE

Figiure 2

Trapezoid characteristic function represented by the template protocol

The development of this method of storing and using fuzzy sets allows for the defining of fuzzy sets directly over the appropriate domains of discourse. Hence, the problem of finding the appropriate values of the input and output gains that would otherwise be required is removed. However, the definition of the appropriate fujssy sets and rules is still critical. In the software developed during the investigation, tlie template parameters were stored in a data base, termed the Fuzzy Set D a t a Base (FSDB). This data base consists of an area of memory where the parameters are held together with a process to manage this aiea of memory and to execute the appropriate read and write operations. Consumer processes request data from the management process, which replies with the requisit data. The relationship between the FSDB and a consumer processes is illustrated by Figure 3, where a process is represented by an ellipse and the area of memory holding the parameters by a tank. The linking of each process is by the point to point communication channels [10] supported by the Occam language, which are shown as arrowed lines crossed by the symbol L This representation is similar to that defined by MASCOT [11], a real-time software design methodology. For the system

424

V ^^IW.fl^|f A^fj:-^

Fym mi

Figure 3

The Stracture of the Fii?:zy Set Data Base

Joint angie/radians

150 200 250 Sample interval "Triangular shapes --Rectangular shapes Figure 4

Gomparision of joint responses produced by firaiy logic controUers using triangular amd rectangular output fuzzy sets

described, it was found by experiment that the shape of any output fuizy set is idatiYely unimportant and hence it was decided to use rectangular output fuiiy sets. This is illustrated by the graphs presented in Figures 4 and 5 which compares the effects of using rectangular aiid triangular diaracteristie output sets. The results displayed in Figures 4 and 5 indicate that although two controllers may differ in their internal structuie they can still behave similarly. This is an area worthy of further investigation as such differences can have signiicant affects upon the computational efficiency of the controller, it is always preferable to nse computationally efficient algorthms in all real time control applications.

3.2

Fuzzy Logic Controller Data Bases

The properties of the diaracteristie functions employed allows for a unique ordering and hence labelling of these sets. Labels [1] which are descriptive of the relationship between faaf sets can be effectively used in the analysis and implementation of a fuzzy logic controller. In common with general practice in control engineering, all fuzzy sets are normalised [12|. The scheme adopted involves specifying a reference point on each fuzzy set and defning a ranking for each of these points. The position of a point in the ordering scheme becomes the rank of the corresponding fua^y set. The reference point for a fuzzy set A, j = Mef(A), is choeen in the following manner:

Donlral aotion/»oHs

V„.:.;;,,/^-:

: ii\J'

-5 10

' •

"

'

-

.

'

"

"

' -

m -

15

-

'

'

:

50

•

.

100

,:

:

150

" • •

^

200

"

••

250

'"1

m»

350

Sample intervai — Triangylar shapes - - Rectangular shapes Figure 5

Comparision of «>nt«jl actions produced by fuiiy logic controllers using triangulax and rectaiigulac output fuzsy sets

1. Find the largest inembersliip grade, £», presient in A £ U:

m 2. Find the smidletit element yeU such that pA{y) = ". This element in t a m e d the reference point of fumzy set A, Ref{A). This scheme is realiiable because of the operational domains over wbich the fuz^y logic controller operaks. Generalisation to variables of type real is a simple operation, if desired. This method was chosen for three reason: • Ease of implementation. • Computational requirements. • Rigour of approach. E x a m p l e 2 Consider ike three fuzsy sets shown in Figure 6. T*e reference peimto forfiazf sets A, B md C are *, go mi S4 mspecUvely. The erier ani lafcfj of A,B and C with rasped to eack other is 1, S and J. This ordering scheme can lead, in certain circumstances, to some minor anomalies depending upon the nature of the fuzzy sets involved. Any "reasonatle" set of "meta-niles" governing the definition of th« fuszy sets and the rules employed in the controller will not allow for such anomalies, though their presence will still be of no practical importance. From equation 4 it is seen that each rule Mi is built, up from three fuzzy sets Bj,, iiCJS,-, and CUi,. The labels for these fuzzy sets can be written in an ordered triple which then describes the rule. For example, rule Ri m represented by the ordered triple (ii,i2,>s). The fuwiy logic control policy, defined by a number of rule., is then described by an equivalent number of triples. Storing these triples in a structured manner aids the search for rules contributing to the controller output, a more detailed description of wbich is given in Section 3.4. A segment of a typical rule base is illustrated in Figure 7, a cross indicating that no rule is present. This segment of a rule base is equivalent to the following fuzzy conditional atatements, where the subscripts are the labels shown on the appropriate edge of the grid in Figure 7. I F Eo A N D RCEa T H E N CUw E t S E I F El A N D RCEt, T H B N CUa E t S K I F Eo A K D RCEi T H E N CUm E L S E I F El A N D RCBi T H E N Ct/ia E L S E . . .

426

4

8

12 1«

Figure 6

20

24

28

34

DOMAIN OF DISCOURSB

Refeience points of some fuzzy sets

ERROR FUZZY SET LABELS

0

1

Figure 7

3.3

2

3

Segment of a rmle base

Defuzziication Processes

Three diiferent defusziflcation strategies were investijsated: the meao of maxima defuMificatioii process, OFMOMi-}; t i e centre of moments deftiziifleation process, DFCOM(.) md a novel strategy, the centre of gravity defuzzilcation process, DFcoai-). The mean of maxima defuzziication process tales the mean of all those elements with maximum membership value in FC{k). It there are m such elements, which are for notationai purposM relabelled a ( l ) , , . . , a(m), the value of w(t) is given by: u(m)

=

=

DFMOM{FC(k})

1 " Z-I]l*FC(i)(e«(«))

(10)

The centre of moments defuziification process usee all the membership grades of FC(k) in calculating the deteimistic cotttrol output. If t i e output fuzzy set is thought of as a two dimensional shape, the deterministic output produced

427

by DFCOM(.) 18 the centroid of the shape produced by the action of the maximum operator on the outputs of the individual rules and is defined by the expression:

The advantages and disadvantages of the previous two methods are well documented [13]. The centre of gravity defuzzification strategy is a novel defuzzification strategy developed for this project. Its difference lies in the use it makes of the outputs from the individual rules and manner in which u){k) is calculated. The DFcoAti) and DFnfOM{-) defuzzification strategies wilt allow the output from a single "rogue '* rule to affect the deterministic output in a disproportionate manner. This results because a single rule can drastically alter the shape of FC{k) and hence the value of uf{k), see equations 4, 7 and 8. A better approach is to calculate the deterministic output in a manner which places more reliance upon a consensus between rules. This approach reduces the effect a single rule can have on the value of Li;(jb). The centre of gravity defuzzification strategy achieves this by using the output from each rule and not their product under the maximum op«rator, FC{k). The output u>{k) is the centroid of the individaul outputs of the rules and not the centroid of FC(k). Improved computational efficiency is an additional benefit since the deterministic output can be calculated directly from the inputs to the controller and the fuzzy set templates. Analytical expressions can be easily determined and two examples of which are presented. This defuzzification technique has three important advantages over the methods more frequently applied: • Ease of computation. • More representative of the decision reached by individual rules. • More robust in the presence of badly chosen rules. Example 3 Lei u(ib) € Ucu (^ ihe deterministic change in control action at sample interval k and the moment of the output for rule /J, be m , ( i ) , as illustrated in Figure 8. let the p contributory ruUa be labelled Ri,..., Rp. The moment of the output with respect to ui(ir) is m,(ik). Hence, for all the rules by definition ofu{k) as the centroid: £"».(*) = 0

(12)

For the Hi, the value of mi(k) can be calculated. For the output shown in Figure 8, a rectangular output fuzzy set: m<(i) = ( 6 i - a O A < ( t ) [ a , ( i ) - ^ i ± ^ ]

Nence j^C'

' "OA-C*) [«(*) - ^ 4 " ^ ] = 0

nJl.(*) = iM^ilMJ)

(13)

(14)

05)

Example 4 / / the ontput fviiy sets are equatatenl trianglet and the fuzzy sets art defined by their width (2li) and the mid-point of their base (at), see Figure 9, it is seen that: hi = h[l-

hi(k)]

(16)

The moment of the output from rule Ri, FCi(k), with respect to u ( t ) is given by: ,r„(k) = IMk) [2 - A.(*)] [w(t) - aj]

(17)

As previously, it is required that for the moments of the p contributory rules with respect to u(k) be zero:

J2m,{k) = 0 i=l P

Hence 5]|/.7.i(it)[2-Ai(t)] K * ) - a,J = 0

(18)

428 MEUBEnSHIP GWOE

DOMAIN OF DISCOURSE

m.m Figure 8

Centre of gravity defuszificafcion strategy for rectangular output fussy sets

MEMBERSHIP GFIADE

h,-4-.

L

^ wik) DOMAIN OF DISCOURSE

Figure 9

Centre of gravity defuszification strategy for triangular oMtput fuzzy sets

429 It can be seen fiom the two previous examples that the expression derived for u(k) can be updated as eaeli contributory rule is discovered. T h b explains the improved computational efficiency of the novel defuziification strategy. The DFcoci-} is far less sensitive to the presence of a rogue rule, and can in some ciicumstances behave in a manner similiar to the DFcomi-), as illustrated hj Figures 10 and 11, although some divergence in behaviour will ol>viouBly ooaur as the nmmber of contributory rules increase. The calculation time for the DFcoai-} is not affected by the shape of the output fujiy sets when the template protocol is used, as indicated by the results presented in Table 1. It is possible to derive expressions for the determini»tic output for other output ckaiacteristic function shapes if so desired. Joint slala/radians 0.05

10

20

30

40

50

60

70

80

«

100

Sample intareal —Centre of moments - - C e n t r e of gramty

Flguee 10

ComparisioD of joint states produced by a fuzzy logic controller using DFCOGQ DFCOM(^)

FC(k) Update tediiuque Modal evaluation timc/ms Minimum evaluation time/ms Maximum evaluatioo time/ms Range in evaluation times/ms Table I

TViajigular output fuzzy sets 0.64 0..51 1.1,5 0.61

Rectangular output fuzzy sets 0.64 0.51 1.09 0.58

Comparision of evaluaiion times using rectangular and triangular output fuszy sete

and

430 Control action/¥olls

10

20

30

40

50

60

Sampi9 inlereal — Canira of m o m m t t - - Cenlre of grawlf

F i g u r e 11

Comparisioii of control actionH produced by fnszy logic comtfoller using DFcoaQ

and

DFCOMQ

3.4

Search for Contributory Rules

Wliate¥ei the process of defuzzification, only a relatively small number of the riiku in the RDB actually make a coBtrlbution to the shape of FC{k) and hence the value of w(Jt). For any given inpat, less than 2.8% (4 rules) and BometiiDies less than 1% (1 rule) of the rule base contribute to the value of w ( t ) . It can be seen from equation 6 that for a rule «,- with inputs e(l:) and rce(l;), if hi(k) = 0 then that rule cannot contribute to the output, whatever defusszification strategy is adopted. This forms the basis of the "iatelligenf rule base search by the author to improve the operational efficiency of the controller. Three rule search processes have been investigated to determine the importance on controller performance of the search method. The irst, termed the Full Search Process (FSP), seeks out valid rules in the RDB for evaluation. The second search process, the Step Search Process (SSP), is more "intelligent" and uses knowledge of the structure of the rules and the rule base to isolate inappropriate segments of the RDB which cannot contribute to the final output. The final search process implemented, a Hippie Search Process (HSP), uijes knowledge of the rale structure, rule data base and the previous contributory rules, A further operational improvement to the RSP is made by commencing the current search at the location of the last contributory rule in the previous interval. The results show not only is t i e calculation time for a fuzzy logic controller less than that achieved with the other search processes, see Tables 3 and 4, the search time using the BSP is relatively unaffected by the size of the RDB. The FSP and SSP calculation times are variable with size of the RDB and number of contributory rules present because of the need to search the RDB for contributory rules. The different rule searches are demonstrated in the sequence of illustrations given in Figures 12, 14 and 16 for a hypothetical rule base made up using 13 fuzzy sets and containing 42 rules. Each square contains the memberahip grades for the input e(l:) and ree(k), for the appropriate fuBy sets. The a»rowed dashed line and shaded region indicates the directiom taken by each search proeea. The starting point for each search process is identified by a "bullet". For the RSP the search starts from one of the previous contributory rules and can change from one search to the next. The FSP increments sequentially throught the EDB and sends to the Fuzzy logic Control Engine (FLCE), which executes the various fuzzy logic operations, all the rules prisent for evaluation. This results in redundant calculations being made and the addditional time penalty resulting from the aissociated data requests and retrivals from the PSDB by the FLCE. The flow diagram for this search process is given in Figure 13, the variables identiied by E . P T B . and B C E . P T B . hold the position of the current location in the RDB under investigation. For example, in Figure 16 for the RSP the starting point of the rule search process h E . P T R = 2 and IICE.PTH. = 4. Figure 12 shows the path taken by the FSP in a rule base for a typical input. In this example, all 42 rules present in the RDB and the 13 fuzzy set membership grades arc evaluated. The SSP, illustrated by the flow diagram in Figure 15, uses the ordering of the rules to search the RDB in a specified manner. If an input is linked with a lero

431 HATE OF CHANGE IN ERROR fUZa SET LABELS

I

I

I

I

I

R3

Is

F i g u r e 12

F i g u r e 13

Example of the FSP

Flow diagram for the FSP

432 RATE OF CWW6E !N BHROH FUZTf SET LAKLS

F i g u r e 14

Example of the SSP

membetsiip grade for a fmsiisy set, the rules employing that fuazy set are not evaluated because the output fuzzy sets will have meiiil>e»hip grades that are all equal to zero. For example, enppcBe the input to the controller at sample period k is: Rom equation 6, the output of rule IJ,- is given by: WftCtjC") = A ['»4*).l*CD-.,(«)] Veil € Ucu Suppose tliat: !*«(,(«(*)) = 0 where t'l = 2 Then fn(jt) = » Hence, |Jrc,(*)(<=") = 0 V CM 6 Ua) Then if foi some other rule B j , jEj, = Si it cam be seen that hj{k) = 0, hence: ffFCiityiev) = 0 V c« e Ucv That is, rule Rj will make no contribution to the output. Tliis will be true of amy other rule using the fuzzy set ,i?j,. A similiar arguetnent will hold for the fnj«y sets deined over UHCE- T M " explanation illustrates why the SSP will mo¥e to seaidi the rules in the next row or column, dependant upon whether the fuzzy error set or the fuzzy rate of change in error set haw a zero membership grade for the current input. The SSP anchor point is static and has been set to the (0,0) location in the EDH. For this example, twelve fuzzy set membership grades were evaluated by the SSP during the search of the B B l . Definition 1 Tie jearei will ie stopped for one of three reasons; 1. The &DB has leen /til%

sarchei.

S. The wmsimum number o/ c&nirikutory rules hum hee.n found. This uumher can be determined 6f the number of input fnzzf sets which hare a n&n~iero membership §rtiie f&r the inpui and the structure of the rslcs. S. A nBinier of comtribsiory rules, less than the mazimvm, further coniributoTf rules.

have ieeii found with no pomiUlitf ef fimdinf any

433

J Figuire 15

Flow diagram for the SSP

RATE OF CHUMGE IN ERROR FUZZf SET UBEIS

3)1 p-°) I io.a)

IttO)

_m_ P.e>

F i g u r e 16

jp.0.7} (0,0.31

Example of the RSP

434

Tke first coniHion is satisfied what all of the rsks havt teea tetied; a properly mhviucei as a defavH stop conditim. Far ofif fttrtieatar infvi there tj a n»aj;ifn»in M%mher of cemtriisioTy raks defined I j the order of the cantmller and the iefnzzificaiion process. When the maximim is reached the search is terminaiei. The mnderlj/mi reasoning iehini lit third coniiliou is more cempki and results from the, relationship hetween fuzzy sets. Let e(k) € Ue ie an error input to ike conlraller and E„ £ UE be the first fszzjj set defined ever the errvr domain that has a non-zero membership §rmie for e.{k). Then any other fsziy set, ssf E'l £ UE, which has a noa-zero memierakip grade for e{k) will be "clo.se' to E„; thai is, tis rank I is 'similiar' to w. Let A he a /sizf set defined ever a rfotnain 0/ discourse U. Then the number of ampperl sets which iBtersect Supp{A) is denoted Jf S,(A). E x a m p l e 5 Consider the fuzif sets illustmted by Pignre 17 and iepnei omr the error domsin; thef are typical of those in the controller developed in this project. For the input e(k) = 13, liE,ieih)) = 0.5, and Si{Ei) = 2 for the fnzzg error sets shown, it is only necessarg to evahaie the membership grades of ike .^ets Ei,E2,E4 and E§; a rtsslt dieieiei ig the order of the fmzzji logic coniroUer.

MEMBERSHIP G R » E

8

« 10 1 ^

16

22 24 26 28

I

F i g u r e 17

32

36 38

42

DOMAIN OF DISCOURSE

Deinition of S/(-)

The ESP is the most sophisticated of the three searches irnnlemented. The approach utilizes the B D l ordering in a manaei similar to the SSE The additional knowledge, however, of the location of past contributory rnlea is also employed to locate an anchor point from which to start the search for the next group of contributory rales. A default anchor point position is available when no previous contributory rules exist, this occurs during the first sample period of the controller. In addition to employing this extra information, the actual pattern of the search is different. The search does not step through rows or columns of the EDB but ripples out from the anchor point, fust searching for non-zero membership grades close to the anchor point, llien progressing to the more distant fuzzy sets. The iow diagram given in Figure 18 illustrates the operation of the RSP and Figure 19 gives a more detailed description of the stop condition. Sets that have labels smaller than that the corresponding set at the anchor point are described as to the left and those to the rif Al have higher valued labels. Separate software roiitiiiei have been developed to search to the left and to the rigM. The operation of tlie routine M G H T . S T E P is shown in Figure 20 and the structure of the l E F T - S T E P routine is similiar. The C O N T I N U E process tests the stop conditions illustrated in the iow diagram Figure 17. The stimulus for a change in search pattern came from the experience gained with the SSP. This experience indicated that contributory rules in successive sample periods are bnilt up using the same or neighbouring fazzy sets. The labels for the fuszy sets used in deining a typical sequence of contributory rules, generated by the controller incorporating the RSP, are shown in Table 2. Starting from an anchor point, the HSP looks for error sets that have a Eon-xero membership grade for the input, the RSP Irst testing the membership grade of the error set at the anchor point and then on either side of this location. Pollowing this, the RSP evaluates membership grades of sets more distant from the anchor point. This process is illustrated by Figure 21 on a typical collection of fuzzy sets. In

435

Figure IS

Flow diagram for the RSP

,TBU6

^

TRUE

Figure 19

Flow diagram for the ftSP stop condition

436

F i g u r e 20 Sample interval 838 839 840 841

Table 2

The Iliglit.step search teat

E fuzzy set label 12 12

n

12 11 12 12 11 11

RCE tuay snot label 1 1 1 1 1 1 2 1 2

CIJ fuzzy set label 7 7 8 7 9 7 7 9 8

Constituent fussy sets for conwjcutive groiipe of contributory rule^

step 1 the aachof point fussy set (C) is t ^ t e d for a oon-sero membership grade for the input. Following that, the search process jumps left, in step 2 and then tigh-i in step 3. Having found a fuKzy set with a non-sero membership grade for the input, the mMch proems continues the search, for any other non-^ero membership grades. Due tc> the ordering of the fuzsy sets^ the search continues by jumping to the right again. The EDB search is terminated using the same stop criteria as outlined for the SSP. The examples given show the operation of the different rule search processes and illustrate that each proems km a dramatically different operation time of eaacli process. In Figuras 12j 14 and 16 the aumber of r u l « which ar<j evaluated are 42, 12 and 6 respectively. The ratio of these nurabere aie of the same order of magnitude as the calculation times required for the fusgy logic controllei incorporating these different search strategies^ the rule h a ^ remained the same in each CIK©. The r^iilts given in Tables S and 4 were generated WiSing two rule ba*^ containing, one containing 81 and the other 289 rules, Prraented in the tabira are the modal, minimum and maximal values of the controller calculation periods for combinatioiia of rule search and rule base sise. The results show that the FSP is the least efficient of the thrM! search processes, the SSP is more efficient and the RSP is the most efficient with the least variation in calculation kimm; the benefit to be gained from using a more sophisticated gearch strategy. The evaluation time of the fuzsy logic controller with the ESP is shown to be relatively unafffi«;ted by the sise of the rule data base. One property of the ESP is that the time spent searching the EDB will be relatively independant of the number

437

DOMAIN OF MSCOUHSe

F i g u r e 21

The BSP searching through neighbomrimg fuzzy sets

of rules or sets prewJiit m the fuzzy logic controller. This results ia the fiizgj logic controller haYing a relatively constant evaluation time whatever the size of the BDB, an important property in the application to the control of dynamic systems. This property will be of added importance in seltorganising fuisy logic controllers where the number of rules apd fumj sets can change. Rule Search Process

rsp

SSP

RSP

Modal evaluation time/ms Miriimuin evaluation time/ms Maximum evaluation time/ms Eaiige in evaluation time«/ms

13.25 13.25 13.5 0.25

1.47 1.41 2.24 0.83

0.64 0.64 0.83 0.19

Table 3

Search procesi evaluation times for a 81 element rule base

Rule Search Process Modal evaJuatioo, time/nis Miiiirniiin evaluation tiine/ms MaAiiimm evaluaiion time/ms Range in evaluation t=im«»/ms Table 4

3.S

FSP 39.74 39.S6 40.51 1.15

SSP 2.94 1.98 2.94 0.96

ESP 0.51 0.51 0.96 0.45

Search process evaluation times for a 289 element rule baae

C o m p e n B a t i o n for A c t u a t o r D y n a m i c s

The Fuzzy logic Controller (FLC) so far discussed has e(ilr) and rce(i;) as inputs, hence it is not capable of compensating for the characteristics of the robot joint actuator. A typical speed-voltage curve for a robot dc-servo motor actutor is given in Figure 22. The dead-uone represents upto 20% of the total voltage range [16] and can adve»ly affect the system response if not compensated for by the control policy. The use of fuzzy logic offers two distinct methods for introdacing the necessary compensation. In the extended fusay logic controller, the individual rules of the fuzsy logic controller are altered to take into consideration the characteristics of the joint actuator. The additional variable required for incorporation into the rules is the value of the manipulated control variable c(k). The more sophisticated fuzsy conditional statecoents

438 MOTOR VELOCtlY/ RADIANS PER SECOND

VOLTAOE/VOLTS

Figure 22

Speed-Voltage Curve for a Typical Joint Actuator

required in this case have been discussed by Zadeh [1]. In this particular case the inputs at the k'th sample interval are e(ib), ri:e(ib) and c(k — 1), the latter being the value of the manipulated control variable in the previous sample interval. The fuzzy conditional statements used are of the form; IF

(error IS POSITIVE SMALL) (rate in change of error IS POSITIVE M E D I U M (current control variable IS NEGATIVE SMALL) (change in control action IS NEGATIVE MEDIUM) where,

AND AND THEN

POSITIVE.BIG S UE POSITIVE-MEDIUM £ URCE NEGATIVE.SMALL £ Uc NEGATIVE.MEDIUM £ Uc

Using previously introduced notation, the characteristic function of the output from the i'th rule is defined by: fFCik) = A ['•((*). W[/,,(c«),/ict/„(cii(t - 1)))

(20)

An alternative approach is to divide the fuzzy logic control policy and the corresponding calculations into two separate processes or components, creating a distributed fuzzy logic controller (DFLC). This approach is based upon the chain of reasoning, discussed by Zadeh [1], with each component equivalent to a block of reasoning within the chain. This form of controller has been termed distributed fuzzy logic controller, since the control policy for each joint is computed by two distinct algorithms. The first of these calculates, or reasons, from an initial premise based on knowledge available. This knowledge is stored in the rules used by this process. The initial premise consists of the values of e{k) and rce{k); the resulting output is a suggested change in control action (^/(ib)) which is used to produce the suggested new manipulated control variable C(i) and cui(k), where: c/(i) = c(t - 1) + cu/(t) (21) The second of these processes continues the reasoning using as additional knowledge the known behaviour of the joint-actuator shown in Figure 22. The input, or premise, for this second block is the output of the first process. The fuzzy conditional statements in the second component of the DFLC have the form: IF ( ci(k) IS SMALL ) AND ( cu/(i) IS POSITIVE SMALL ) THEN ( cu(t) IS POSITIVE M E D I U M )

439

The diitributed controUei can now be seen to consist of two sets of fnsisy conditioaal statements. This method allows complex control policies to be implemented by a series of simpler processes. This apptoach conformi to the Zafcdeh [1] idea of how the human mind operates, An initial premise, generated from an input, is modifled hj later reasoning based upon additional information.

3.6

Implementation of the Distributed Controller

Since the first component of the D F i C already exist in the form of the FLC, it is only necessary to implement the second component. The second component consists of fuzzy conditional statements wbieh implement a control policy dictated by the cliaracteristics of the joint actuator shown by Figure 22. The aim of the second component of the DFLC is to compensate for the dead zone shown in the speed-Yoltage characteristic. Tkis is achieved by ensuring the final value of the manipulated control variable to ensure that it passes through the actuator dead none, notationally this is rpresented by the closed intetwl in the control variable domain [D<„ Dj] C Uc, in the minimam possible time. For example, suppose at some sample interval k that c{k} is within the dead Sione and u(k} is positive, one possible cause of action would be to assign c(jfc) the maidmum value of the dead gone, plus 3 volts. A control policy utilizing a variable output gain will adequately compensate for the dead-zone, hence the fuuy conditional statement can be replaced by » btancMng statement under the control of a Boolean « 3.6.1

C o n t r o l of Variable C h a t t e r

In the initial implementation of this component, the manipulated control variable was switched directly across the dead zone using the following algorithm: IF[c(l;) 6 [D^.Di A N D w(t) > 0 ] T H E N c ( t ) = Dj E L S E I F lc(k) € [£>., O J A N D u{k) < 0] THEMc(l:) = U . It was found experimentally that this approach led to rapid changes in the manipulated control variable "chatter* aa the joint approadied the steady state value. An example is given in Figure 23, where the joint state and manipulated control variable value, are plotted on the f-axis with sample interval along the i-axis. Joint angle/fKlians

Control action/votis

; : .- ||

0.3

.

•-.;; 7 ; / i : - - --/.;,;. ,. , -,• - . - - . / . ^ i

0,2

-••

- /

'-'

' - "

..:

•'

• . • - . . ; „ • ;

: ; • •

,

:•'•

,:

"•'

•-

•':

. ' " : ' -

•

. . - . , • .

t

HWIBWBMIBMMiBHI

0

1C»

2CM

300

400

5CMJ

600

700

Sample interval — Joint angls/radians --Control variatile/volts

F i g u r e 23

Chatter in the second component of the D F t C

These results were generated from an input demand to the shonlder joint of 0.343 radians. An oscillation of ±0.07 radians around tlie desired value, with a period of approximately 0.5 seconds, can be observed in the data displayed in Figure 23. For negative voltages the output torque from the motor will oppose the gravitational force exerted on the shoulder joint which explains the negative bias in the values of the manipulated control variable. The

440

dead band has been highlighted in the graph, the zone between -3.? volts and +4 volts, where shatp changes in the manipulated control variable can be observed due to the action of the second component of the DFLC. The problem of chatter was overcome by using the l i s t component of the DFLC to alter the value of c(i) IB the small zone around 0 volts, [Di„J?J C Uc- By experimenl., the authors discovered that Di = - 0 . 8 volts and De = +0.6 volts were adequate for all joints. The benefit of this approach can be obseived in Figure 25, wlieie the conditions remain the same x for Figmr« 23. It should be noted the changes in value of e{k) are both smaller in magnitu
K g e p e 24

3.7

Flow diagram representation of the control policy of the second component in the DFLC

The Hybrid

Controller

The Hybrid Controller (HC) consists of three components. The ftiziy logic controller which caa be either the basic or distributed version, which operates over the domain UB- The second component is a standard deterministic PID controller which operates over a small subset of UE and inally the supervisory process. This latter process dictates which component of the HG is active at any particular sample interval. A schematic representation of the HC is given by Figure 26. The components of the hybrid controller operate in different regions of UE • Experience gained in the use of the FLC and DFLC indicates improved dynamic behaviour is possible in the steady state region ( ±0.2% = (±0.012 radians) ) if the requirement exists. The supervisory process activates the d«iied component, either the fuzzy or determinisitic controller element and the activated control process, euFUZE(t} or ctiDErit) is used to update the manipulated control variable, c(k). The default values for eafvzglk) and caDBrfi) are aero, these are altered only if the appropriate component of the distributed controller is activated and calculates a value other than leio for either ctiji-i/zz(i) « eUDBT(k)- Hence,

441

the value of e(i') is given by: e(k) = 4*= - 1) + cuFi/zzC*) + euDKrC*) (22) The program stmctnre of the hybrid controller is ilhstratexl by Figure 27, The authora chose to investigate a Control acliDn/¥oKs

.loint anglo/radiana

25

0.4 0.35

20

^

0.3

^15 ^10

0.25 0.2

5 Q.15 0

0.1 O.CS 0

•" /I .\ '• . l\ '', f

L_l^

0

"

-5 .10

50 1C»150 2 » 2 5 0

300350400450S<MS50

SamplB int#real — Joint angto/radians —Control wariaMs/voltss

F i g u r e 25

Reduced cliatter in tlie rnodified second component of the D F i C

„(•'>

DEMAND d|k)|

«(m ,

L.»»-j

STATE t W

=%f*)

CONTROLIEF INPUT

COirmOL ACTION

F i g u r e 26

Schematic representation of tlie hybrid controller

l i e that comprises ti»e DFLC and PID components because of the lesulte obtained earlier in tlie project. It is interesting to note that during the development phase of the of the HC, both PD and PI algotithme were used as the deterministic controller components and the results showed little difference in lesponse. The FLG omtlined earlier does not require any modiication to be made and can be simply used directly in the hybrid controller. Similiaily, the control policy executed by the DFLC can be left undianged. There is, however, one major dificulty. In operation it was found that a HC containing the DFLC as the (mif controller component will experience relatively large "kicks* in the value of c{je) as the system approaches the steady state value and this

442

UPDATE.DQNWtOC AC 1K>N

K g n i « 27

Pseudo code and flow diagram for the supervisory process

leads to oscillatory uiotion of umaeceptably large amplitude. This b e h a w u r is moat obvious in the waist joint. The magmitiide and frequency at the waist joint ia illustrated by the dashed lime in Figure 28. This behaionr, however, is BOt as pronounced in the remaining robot joints and can be attributed to the low reaction torgue of the waist joint actuator as the Joint reaches the selected point. There are two possible solutions to this problem. Joint angto/radians

80

100

120

140

150

180

2C»

SamplB intarval — Improved DFLC --UnaJtered DFLC

F i g u r e 28

Conflict b«tweeB components of the EC

The irst involves tcsttncturing the hybrid controller so botli control components are active together with adjustable "weights" on each component output value. The value of the manipulated variable will then be obtained from the sum of these components. Suppose GFazz(4>')} " d GDMTie(k)} are the chosen weights, as shown in Figure 29,

443 MEMBERSHIP QRAOE

ERROR DOMAIN/RADIANS

CHANQE IN CONTROL ACTION GAIN FOR DCTERMINISTIC CONTROLLER CHANGE IN CONTROL ACTION GAIN FOR FUZZY LOGIC CONTROLLER

F i g u r e 29

Variable gain values for the hybrid controller

then the value of the manipulated control variable is defined by: c(t) = <:(* - 1) + [GFi,zz(e(i)) x curuzz(.k)] + [GoErMk))

x CUDETC*)]

(23)

The second solution, which was adopted, is computationally less demanding and more elegant. In this latter approach, the various components of the distributed fuzzy logic controller are switched off in a predefined sequence as e{k) tends to 0. It was found by trial and observation that if the second component of the distributed fuzzy logic controller was deactivated when the joint luigle B{k) was within the range of ±0.004 radians of the desired value, the previously observed *'kicks^ did not occur. The result of using this approach is shown by the continous line in Figure 29. It was assumed that the robot joint could be modelled adequately using a second order equation, since similiar models have been used successfully [14]. The assumption is reasonable in this particular case because the nnodel only represents the behaviour of the joint in a small region around the steady state value. The model properties were evaluated using a technique based upon the logorithmic decrement application to under-damped system [15], and were estimated to be ( = 0.6 and w„ = 5.0. Using these values, an initial set of PID controller coflicients were derived. By trial and observation these values were tuned to the robot, yielding: (24)

iCd = O.I, /Ci = 2,0, /Cp = 2.5

The PID velocity controller was implemented, both for safety and for bumpless transfer. The problems encountered in implementation were created by the digital feedback signal from the optical encoder and the limitations of the Occam language which introduces rounding errors. Both these factors reduced the stability margin. It was further noticed that the performance of the velocity algorithm resulted in different outputs depending on the number of rounding or truncation operations that occur. The problems that arise in the digital implementation of algorithms when dealing with rounding errors and variable sample rates is discussed by Moroney [16] and an explanation is given for the relatively poor response of digital controllers. The expression used to implement the velocity PID algorithm use in the HC is:

-'(*) = t.

(e(t) - 2e(t - 1) + e(k - 2)] + K, [e(J:) - e(t - 1)] +

Kie{k)t.

(25)

The supervisory process of the hybrid controller consists of two routines and these are error driven. The first routine switches between the DFLC and the PID components of the HC using the boundary values discussed earlier. The second routine ensures both of the components ( DFLC and HC ) cannot be evaluated together.

444 jQ'mi angle/radians

0

?0

40

60

80

100

1»

140

160 180 ?00

Sample Intsfvat

— SFLC " - H 0

Figiiice 30

Conipariaion of the behaviour of the FLC and HC to s step demand

Figure 30 shows the waist joint responses for both the FLC and HC for a step dieturbance of 0,0343 radians. The FLC has a superior transient response to thai of the DFLC, wliich lias a noticable and also a better steady itate performance. In addition, the HC exhibits a greater overshoot and increasing the damping parameter value in the PID component of the HC results in a longer rise time with a larger steady itate error. Figure 31 shows typical responses for the shoulder joint when under the control of either the FLC ot the HC; the perturbation was caused by the motion of the elbow joint. Joint angle/radians

150

200

2S0

Sampla interval — SFLC ^-HFLC

Figure 31

Comparision of the beliaviout of the shoulder joint in the presence of a disturbance

It should be noted that all the experimental results were obtained using the lame data bases were appK)priate,It can be seen that the response of the joint is superior when the FLC algorithm is employed. Not only ate the

oBcillations experienced by the shoulder joint smaller, the steady state response is much reduced. It can be seen from these results that the performance of the "pure" fuzzy controllers ia superior to the hybrid controller. This result confirms that the fuzzy controllers can respond more positively to both digital input data and externa] perturbations.

4

Project Software

The experimental investigation of the proposed controllers required the development of appropriate software. The origianl design problem was decomposed into a number of simpler tasks before coding was undertakene. The design goals were: • Control the major robot joints using fuzzy logic. • GBUS board to be operated by a distinct process. • Simple interface between the controller software, user and host computer filer system. • Modular structure, provision for the upgrading of the individual units comprising the software package.

4.1

Top Level Design

The algorithms used, reviewed earlier, form the core of the controllers implemented and studied. requirements enabled several distinct features to be identified:

The design

• User interface. This facility allows the controller to be initialized and modifications to be made during operation. • Interface from the transputer system to the host computer system, an INTEL 80486 based personal computer. This enables the control software to access the data bases and store run time data on the hard disc. • Process to control the GBUS interface card. • Joint coordinate process. Includes a simple path planner and inverse kinematic solution process. • Joint controllers. • Collision detection. Actuator overload detection for each joint. An "engineering approach" was adopted to the design of the concurrent control software. A combination of techniques were adopted, such as a top-down design methodology using MASCOT like primitives, flow diagrams and pseudo code. The approach to testing is bottom-up, allowing for the resolution of communication deadlocks and other common problems common to Occam software. A discussion of these techiques is presented by Henderson [18].

4.2

Mapping Software Processes onto the lYansputer Network

The most effective use of a network occurs when individual transputers require the minimum of communications with one another. Figure 32 displays the mapping employed for the processes used in this project. This configuration was found by experimentaticm to be the most efficient. A number of alternative arrangements were tested which were similiar and most likely to achieve the same desired goal.

4.3

Controller Input and Output Processes

Input and output processes are required to support the operation of the controller. The input process provides data in the appropriate form to the controller and the output process examines the output from the controller to identify possible collisions or actuator overloads. Each joint controller is supported by a dedicated input and output process. The input process calculates e{k) and rce{k) from the difference in desired and actual joint states as described in Section 4. The desired joint state, or demand, is available from the control interface process and the current joint values from the GBUS card.

446

HOST T800

l TFIAMSPLrrER

TFWWSPUIEB

F i g u r e 32

4.4

Mapping Processes onto the Transputer Network

Internal Structure of the Fuzzy Logic Controller

The basic fuasssy logic coiitroller is made up of five processes: 1. Puszy logic eagine. 2. Kmle seaich process. 3. Rule data base. 4. Fuziy s«t data base. 5. Data base inter-relatioEal procesii. The structure of the basic fuzzy logic contiolkt is illustrated in Figure 33, The moie sophisticated D F i C and hybrid controllers hme addional processes, dependant upon the additional complexity of the control policy. They are discussed in a later section. The fuzzy logic engine i« the centre of the controller, its exact form of implementation is dependant upon the rule search and defmziification processes used. Rules ate presented by the rule search procesB for evaluation to the fuizy engine which can then calculate the deterministic output. The basic rule search pioeese sends all rules to the fuzzy logic engine while the more complex search processes send contributory rules for evaluation. The sophisticated rule search processes partially evaluate the rules in their search and to avoid duplicating calcualation, partially evaluated rules are sent directly to the fuzzy logic engine together with the identity of contributory tules.

4.5

Operations upon the Data Base

The fuzzy set data base stores the template parameters used by the fuzzy logic engine to evaluate rules. Because dynanjic arrays are not available in Occam, the template parameters ate stored in faed size arrays. Control characters are also included in these array, to indicate whether particular segments of the array hold valid template parameters. The arrays used were of type [N][5]!ST where N is the maximum number of fuzzy sets used, typically 17. The array is of width 8, to store the 4 trapezoid template parameters and a control character. For a single joint three such arrays ate neccessary, one for each of the fuzzy sets deflned over UE, UKCE and UcuA management system executes all read and write operations on the arrays, processes wishing to interogate the data base do so through the managament system and in addition also carries out all the required data base "house

FROM CONTROL . INTERFACE - T

FROM CONTROL INTERFACE

F i g u r e 33

The Fuzzy Ixsgic Controller Processes

keeping" aclivities. For example, rearranging the data in the anay when new template parameters are added or deleted. It ensures that the data is held contiguoysly, an airangtnent which will simplify the operatioE of the data base and lience improve the run time efficiency of the program. The template paramteis within the data base are stored in rank order. The array offset is equal to the ranking of the fuzzy set. The deletion or addition of a fuzzy set can result in changes to the ranking of the other fuzisy sets. To maintain contiguous use of the array this implies mo¥iiig template parameters. This action is carried out by the management system which governs the area of memory holding the array. 4.6

Rule Search Process

The rule search processes implemented have been discusaed in detail in Seclioo 3. Their fole is to find the appropriate rules for evaluation. In the simplest rule search process all rules are evaluated. The mofe sophisticated rule search processes identify only those rules which can contribute to the final deterraiiii.'stic output. In these search processes, the rules are partially evaluated during the search an
4.T

The Coordiaation Level

The coordination level is made up of several processes which calculate the appropriate demand values for the local controllers and send these ¥alues to the local joint controllers in a synchronised manner. From the end effector location, an inverse kinematic model is used to calculate the joint angles. 4.8

H y b r i d and Hierarchical Fuzzy Logic Controller

The hierarchical controller is, as previously described, the single layer controller extended to include an additional control phase. The second layer testa for one of several conditions and executes a particular option depending on which condition is T H U E . The hybrid controller is a combination of the hierarchical controller and a deterministic controller. A decision procesSj which activates the appropriate component of the hybrid controller, is required. The relationship between the constituent processes of the hybrid controller is shown in Figure M.

448 FHOM INPUT PRCXESS

^

%

'*i FBOM CONTHOL

vt

^

FROM CONTBOL INTERFACE

TO OirTPUT PROCESS

Figm-e 34

5

Tke Hybrid Controller

Concliision

The results given in this paper show that fiijiy logic can be used in robotic control and that the steady slate problem often encountered in the application of fuisaj- logic control can be overcome through the use of a hybrid controller. A fuzzy logic controller caw be designed to operate in the real time enviromemt with high sampling rates without modiication of the controller's structnre. It has been shown that parallel processing using a transputer network caa be successfully applied to real time control in robotics. The design and development of small to medium siaed programs (8000 lines), written in the language Occam, can readily be cli¥i8e€l for applicatioES requiring real time operation.

References [1] Zadeh, L. A. Outline of a New Approach to the Analysis of Comp-lex Systems and Decision Processes. IEEE. S M C , January 1973, 3(1), 28-34. p ! Daley, S. and Gill, K. F. Attitude control of a spacecraft udng an extended self-organising fuizy logiccontroller. Proc. Int. Eng., 1987, 20(12), 97-106. p ] INMOS htd. The Transputer Developement System. Prentice-IIall, 1988, [4j Poutaiii, D. and May, D. A Tutorial Introduction to Occam Programming. BSP Profeasional Books, 1987, PI Mitsubishi Electrical Coorpoiation 1980. Move Master II: The EM-501 Model Industrial Handbook, [6| Sension l t d . Sension Transputer Products: The TDS Delivery Manual. November, 1988.

This page intentionally blank

[7] Sension Ltd. Sension TVansputer Products: The GBUS96 Delivery Manual. Issue C, February 1989. [8] Mamdani, E. H. Applications of fuzzy algorithms for the control of a dynamic plant. Proc. lEE., 1974, 121(12), 1585-158S. [9] Procyk, T. J. and Mamdani, E. H. A Linguistic Self-Organising Process Controller. Automatica, 1985, 15, 15-30. [10] Andersen T. R. and Nielsen, S. B. An efficient Single Output Fuzzy Control Algorithm for Adaptive Applications. Automatica, 1985, 21(5), 539-549. [11] Bennett, S. An Real Time Computer Control: An Introduction. Prentice-Hatl International (UK) Ltd, 1988. [12] Kaufman, S. and Gupta, M. M. Fuzzy Mathematical Methods in Engineering and Management Science. North-Holland, 1991. [13] Lee, C. C. Fuzzy Logic in Control Systems. IEEE. SMC, 1990, 20(2), 404-431. [14] Fu, K. S., Gonzalez, R. C. and Lee, C. S. G. Robotics; Control, Sensing, Vision and Intelligence. McGraw-Hill International Industrial Engineering Series, 1988. [15] Raven, F. H. Automatic Control Engineering. McGraw-Hill Mechanical Engineering Series, Fourth Edition, 1987. [16] Electro-Craft Ltd. DC Motors, Speed Controls, Servo Systems 5th Edition 1980. [17] Moroney, P. Issues in the Implementation of Digital Feedback Compensation. The MIT Press, 1983. [18] Henderson, M. I. Use of Parallel Processing in the Fuzzy Logic Control of a Robot. Doctoral Thesis, May 1993. The University of Leeds, England.

C H A P T E R 17

FUZZY CONTROL OF ROBOTIC MANIPULATORS AND MECHANICAL SYSTEMS R. Gorez and M. De Neyer Centre for Systems Engineering and Applied Mechanics University of Louvain, Bdlg Euler, Av. G. Lemaitre, 4-6 B-1348 Louvain-La-Neuve, Belgium

1

Introduction

The Robotic Institute of America (R.I.A.) gives the following definition of robots: " A robot is a reprogrammable multifunctional manipulator designed to move material, parts , tools or specialized devices through variable programmed motions for the performance of a variety of tasks" [55]. Based on this definition it is apparent that a robot must be able to operate automatically. This implies that in most of the robots it is possible to distinguish the following major subsystems: a manipulator (mechanical unit which can be compared to the skeleton of living beings), sensors and actuators (sensory organs and muscles of living beings), a controller (the brain), appropriate power supplies, and very often a computer system which takes care of the monitoring and control functions relative to the robot operation and which allows exchange of d a t a between the robot and human operators a n d / o r other parts of the manufacturing process in which the robot is performing some specified tasks. T h e motions of the manipulator must be controlled and the control system obeys the same basic principles as for control of motions of any mechanical system from simple servomecbanisms up to complex machines or vehicles. It implies that positions and velocities or displacements of the various parts of the mechanical system must be monitored and that related data must be transinitted to the control system. Then the latter is able to determine the driving forces a n d / o r torques which must be applied to the mechanical system in order to force the actual positions and displacements to track the desired ones. T h e next section of this chapter is dedicated to the dynamics and control of robotic manipulators as they are relatively complex mechanical systems. Then the basic principles of control of motions in mechanical systems are reviewed in the following section. T h e two major classes of conventional position controllers are introduced in a unifying presentation. This presentation aims at introducing fuzzy control in the following section, fuzzy control appearing then as a natural extension of multilevel discontinuous control. T h e advantages of simplicity and reliability of discontinuous ' T h i s paper present.s research results of the Belgian Programme on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister's Office, Science Policy Programming. The scientific responsibility rests with its authors. 451 S.G. Tzafestas and A.N. Venetsanopoulos (eds.j Fuzzy Reasoning in Information, Decision and Control Systems, 451-491. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.

452

control are retained by fnzzy control, but not its main drawback which is continuous cycling between different types of operation. Therefore fuzzy control can be viewed as an intermediate class between discontinuous and linear control systems, resulting in an acceptable compromise between advantages and drawbacks of both. Positions and velocities or displacements are the usual state variables defining the state of a mechanical system. Control laws based on measurements or estimates of those variables allow some changes in the dynamics of the system, in particular the stabilization of unstable or neutrally stable systems, and give them the capability of reproducing desired motions with an accuracy which depends on the gains of the control system. However in such control systems static errors due to steady-state loading forces cannot be avoided. T h e only way to cope with such disturbances and reject their effect on the system is the introduction of a reset action in the control system. This can be achieved through parallel controllers using control laws basetl on triples instead of pairs of data. However it may result in some deterioration of the dynamic performances and lead to more difficulties in the design of fuzzy controllers. Self-organizing controllers are a possible solution to the latter problem. Nevertheless there is another way since an indirect reset action can be introduced via model-based control schemes. These allow a neat separation of the two basic tasks of the control system: following the desired trajectory (tracking) on one hand and reducing the effect of disturbances (regulation or disturbance rejection) on the other hand. Such control schemes consist of two control loops, one of them including a basic position-fvelocity controller and the other one a model of the system. The basic controller a n d / o r the model can be implemented as numerical or fuzzy systems as it is shown in the fourth section of this chapter. Position or displacement control is not the only type of control which may be required in the performance of robotic tasks. In some of them one has to control contact forces between the robot-end-effector and the robot environment. Often the complete control system of a robotic manipulator is hybrid in the sense that displacements are controlled directly along some directions in the robot workspace and forces along others. Then force control is generally implemented as an external control loop around usual displacement control loops. Here again use of fuzzy logic is possible. Other applications of fuzzy logic in robotics are also mentioned or presented briefly in this chapter, and some paths for future developments of fuzzy control of roliotic manipidators and more generally of mechanical systems are evoked in the concluding section.

2 2.1

Dynamics and control of robotic manipulators Dynamics of a robotic manipulator

A robotic manipulator is a relatively complex mechanical system with a joint-link structure. Basically there is a serial collection of mechanical linkages connected by joints to form an open-loop kinematic chain. The first structural part of the chain is the robot arm; it aims at moving the robot-end-effector through the robot work space and includes the major joint-link pairs, these being generally located closest to the robot base and also the most robust from a mechanical viewpoint. A second part of the chain is the wrist which includes the minor linkages and must give the end-effector

453

fine positioning and the desired orientation. Joints may be revolute joints producing pure rotary or rotational motions or they may be prismatic joints allowing linear or translational relative displacements. Each joint defines a robot axis around or along which there is a relative rotation or translation of the subsequent linkage of the chain with respect to the antecedent one. These axes are usually but improperly called degrees of freedom (DOFs) by engineers. Most of the manipulators fall into one of the following classes; Cartesian, cylindrical, spherical or jointed devices, according to the coordinate system which is appropriate for describing the configuration of the major axes. Fig. 1 shows a six-DOF jointed-arm manipulator. Very often indeed the number of DOFs is five or six, three for positioning and two or three for orientation depending on the tasks the robot is designed for. However the number of DOFs may be as small as four or even two in a few robots; on the contrary it is larger in others which have redundant axes giving the manipulator more flexibility in adapting its own configuration to a particular task. Besides in some applications the base of the robot is carried by a vehicle allowing large linear displacements in at least one direction.

Figure 1: P u m a 760C manipulator (Reproduced by permission of Staubli Unimation)

T h e motion of the robot-end-effector is obtained through coordinated simultaneous displacements along the various robot axes. Then the kinematics of the complete

454 chain and the dynamics of all the manipulator axes must be taken into account. Each link is driven by a joint actuator and has mass and inertia associated with it, the inertia depending on the configuration of the subsequent Hnks. There are also frictional forces in the joint and forces due to the gravity may act upon the link. Besides when several linkages are moving simultaneously their motions are interacting due to Coriolis and centrifugal forces which are exerted on bodies moving with respect to a moving reference frame. All these forces are nonlinear in nature and the dynamics of the complete link chain is complex. However a dynamical model is desirable for the design of the robot control system. Equations of motion of the manipulator can be obtained via a Lagrangian or a Newton-Euler formulation [55, 79]. T h e second formulation has been found to be superior for fast calculations, either for computer simulation or for real-time dynamics computation in some control systems. However Lagrange's equations are in a simple form allowing the analysis of the effects of various parameters on the motion. For this reason it has been a standard approach over the past twenty years [55, 76, 79]. T h e Lagrangian formulation utilizes generalized coordinates and forces; typically, if the links can be assumed to be rigid bodies and if spring effects in the joints can be neglected, generalized coordinates and forces are respectively the relative displacements in the joints and the forces or torques associated to the manipulator axes:

Mi^l) • q = ~9i(l) - /(
+ T,„

(1)

• q is the n-vector of joint displacements, n being the number of robot axes, • q and q are respectively the vectors of joint velocities and accelerations, • T denotes a vector of generalized torques or forces, the subscripts m and / referring to the driving and loading torques respectively, • M{q) and C[q,q) are the inertia matrix of the manipulator and the matrix relating centripetal and Coriolis forces to the generalized velocities, • g{q) and /(
455

2.2

Control of a manipulator

Roughly speaking from a control viewpoint, it can be said that the tasks a robot has to perform fall into two classes: position control and force control [55]. Position or trajectory control is used when the robot-end-effector must be moved with or without load along a prescribed trajectory through the robot workspace. Examples of such tasks are spray painting, welding, parts handling, palettizing, . . . . Only the endpoints of the motion are specified (point-to-point control) in many of those applications, but often the user is allowed specifying also some intermediate points a n d / o r the average velocity of the manipulator. In those applications the trajectory connecting the specified points is free and the robot control system determines itself a velocity program for each robot joint in order to satisfy the above prescriptions. In many industrial robots the user can also prescribe the manipulator to proceed from one point to the following one along a linear or a circular trajectory. In other applications the desired trajectory of the end-effector is completely specified beforehand and positions and velocities corresponding to a sequence of points along the desired trajectory (up to several thousands) are stored in the computer memory (continuous path control). T h e control system of most industrial robots is simply a collection of joint controllers each of which is dedicated to a single joint and drives it individually (Fig. 2). T h e reference signals for these controllers are supplied by a joint-trajectory generator determining the desired joint trajectory from the desired trajectory of the end-effector. Oenerally the joint controller is a feedback controller having two terms respectively proportional to position and velocity errors and introducing proportional and derivative actions (P-D control) or it is a three term feedback controller with proportional, derivative and integral actions (P-I-D control). Besides the controller may include a feedforward term corresponding to the desired acceleration and a feedforward or feedback compensation of the major disturbing torques, in particular those due to gravity. In some cases compensations of the variations of the moment of inertia are also introduced in the controller gain with a view of achieving dynamic performances independent of the robot position and configuration. Other control systems including linear quadratic optimal controllers, adaptive controllers, disturbance observers, variable structure control s y s t e m s , . . . are proposed in the literature; see e.g. [94, 105]. In all the previous applications it was assumed that neither the end-effector nor any other moving part of the manipulator comes in contact with any object in the robot environment. Then the implementation of the control system need only joint position and velocity measurements and some knowledge of a dynamical model of the manipulator. Other tasks such as assembly, deburring, polishing, cooperative work with other robots, . . . require the control of the position and also of some forces exerted by the end-effector upon an object. This can be achieved via two different schemes: impedance control and hybrid control. Impedance control aims at adjusting the mechanical impedance relating the end-effector displacement to the forces due to contact with the environment of the robot. Impedance control may be passive or active. In the first case a kind of elastic suspension including springs and dampers is inserted in the connection between the robot-end-effector and the manipulator wrist. In active impedance control the feedback controller utilizes measurements of position and velocity of the end-effector with respect to the object and measurements of the contact forces between them. A noteworthy difference between position and force

456

desired trajectories driving torques for of joint 1 joint 1 Controller for joint 1 Desired joint trajectory generator

joint 2

joi it n

C'ontroller for joint 2

joint 2

Controller for joint n

joint n

positions and velocitie; of joint

joint 2

Robot joint n

Sensors for joint n

Sensors for joint 2 Sensors for joint 1 Figure 2: Control system of a n - D O F industrial robot control is t h a t position control in general should be very accurate while force control may be looser; usually all that is required is that the actual contact force or torque lies within a given range. Then qualitative control and fuzzy concepts appear to be very natural in that field. Hybrid control includes feedback loops providing separate control of displacements along some directions and forces in others. It should be noted that many of the previous control systems require measurements in the workspace and actuation of the joints; this implies a lot of on-line calculations for transformations from the workspace into the space of joint coordinates and vice-versa. Then some implementations of the control system are two-stage structures including inner loops for joint-position control and outer loops for force, impedance or hybrid control in the robot workspace. For this reason joint position control is the main topics which will be considered in the followings. Thanks to the simplifying assumptions explained in the previous section it is generally sufficient to consider separately each joint, by regarding changes in the manipulator dynamics, joint interactions, friction and gravitational forces and possible loading forces, as external disturbances the position controller of the considered joint has to cope with.

457

3

Control of mechanical systems

With the above assumptions the model relating the applied torques or forces to the position of a one-DOF mechanical system is a double integrator (Fig. 3). The controllers which are the most frequently used for position control of such a system are two- or three-term linear controllers and discontinuous controllers. External Disturbances

Interna 1 Disturbances

\, >'

. Tl

\

,

)

'


r

(1

J

Figure 3: Model of a one-DOF mechanical system

3.1

Linear control

As it can be found in most textbooks on control engineering, see e.g. [19, 74], the basic linear controller for position control of a simple mechanical system is a P-D controller with two terms respectively proportional to position and velocity errors (Fig. 4). Then the state equations of the controlled system are: dq

Tt = ' M.-

=

r„.-r^

(•2)

and the input-output relationships of the controller are:

e-v Tjj^

=

(Id-

—

•'"•p * ^ P

(3)

q f

Ji-t/

where q and q are position and velocity, subscript d refering to the desired position and velocity, Cp and e„ are respectively the position and velocity errors, A'p and A'„ the corresponding controller gains, T„^ and r; are the driving and loading torques respectively and M denotes either the mass or the moment of inertia of the system. T h e closed-loop system can be described by the following second-order differential equation relating the position error e to the desired acceleration and the loading torque: M • E + K^ • e + Kp • e = M • qi + Ti (4)

458

External Disturbances

(Id +

Figure 4: PD control of a mechanical system Setting this equation in the standard form of a second-order linear differential equation: e -I- 2Cw„ • e 4- uj^ ' '^'' ^ 'i'l + ^

"^i

(5)

it is well known that the complementary solution [74] of this differential equation is a linear combination of two independent functions; •

exp{(-C + y e ^ n " ) a ; „ < }

and

•

exp{—Cu;„<} • cos{w„(}

and

•

exp{—Cw,ii}

and

It appears that u;,i

e x p { ( - C - v T ^ " ^ T ) a;„<} in the overdamped case (( > 1), exp{—C'^,,^} • sin{a;„<} in the underdamped case (C < 1), t • exp{~(uj,it} in the critically damped case (C =

determines the response time of the closed-loop system

K

is related to the aperiodic or oscillatory feature of the transients 2Miu„ and to their decay ratio. As indicated by these relationships the characteristics of the dynamics of the closed-loop system can be adjusted independently by a proper choice of the values of the proportional and derivative gains Kp and / \ „ . It is also obvious from Eq. 4 t h a t the steady-state error due to a static loading torque is inversely proportional to the servo stiffness A'p.

and C =

Then the accuracy of the position control system is directly related to the servo stiffness A'p. ffowever it can be improved by adding in the control action an estimate •f( of the loading torque a n d / o r a feedforward term M -qri corresponding to the desired motion {M is an estimate of the inertia of the controlled system). On the contrary the dynamic error could be higher if the derivative term in the control action contains only a velocity feedback — A'„(j instead of the velocity error A'„ • e„.

459 Another way towards the rejection of disturbances, more precisely towards the reduction of errors due to slowly varying disturbances, is the introduction of an integral action in the controller: T,n = Kp • Bp + Ky • e„ + I
Cp dt

(6)

This additional action results in a cancellation of the position error in steady-state; for this reason it is called the reset term in many textbooks on control engineering, see e.g. [74]. It should be noted however that the closed-loop system is now a thirdorder system, which may lead to a possible deterioration of the transient dynamics and to some risks of instability. Nevertheless these problems can be passed around by decoupling the tracking and regulation dynamics through a model-based control scheme as explained further.

3.2

Discontinuous control

Another controller which can be used in some applications of position control is a discontinuous controller with only two or three possible values of the manipulated variable. T h e first controller of that type is the on-off or two-position controller where the control action can take only two values depending on whether the input signal of the controller is positive or negative. In position control of mechanical systems these values are -\-TM and —TM, TM being the maximum value of the driving torque, and the input signal is given by a so-called switching function 5 the arguments of which are the position error and either the velocity error or the actual velocity. Then the binary control law can be expressed by the following very simple mathematical relationship: Tm = TM • signjs} (7) where sign{.} denotes the sign-function: signjus} signjs}

= =

-f 1 if .s > 0 - 1 if s < 0

(8)

It can be seen on Fig. 5 and 6 that the phase plane (e, e) is divided by the switching line s — (] in two parts inside each of which the driving torque is constant. Neglecting the loading torque and other inhomogeneous terms such as the desired acceleration the equation of the closed-loop system can be written down as follows: • de

de

1

.

r i

/„x

and clearly it has the solution:

M e+ In other words, in a positive or and forth in a system being a

• e^ • signj^s} = constant

(10)

due to the fact that a constant driving torque is always applied either in a negative sense, the controlled system is bound to be driven back cyclic manner, the trajectory in the phase plane of the closed-loop sequence of alternating arcs of parabola.

460

0.5-

-0.5

-1.5

Figure 5: Discontinuous control (switching function is optimal for M — I)

0.5 -

-0.5-

Figure 6: Discontinuous control (linear switching) T h e best behaviour t h a t can be achieved is the so-called bang-bang control where

461

the trajectory corresponding to simple cases such as free response or step response is composed of two arcs only, the first one starting from the initial point in the phase plane and the second one passing through the origin of the phase plane. This would be the minimum time response for any driving torque bounded to take its values in the interval {—TM,+TM), as shown in [74]. Then the switching line in the phase plane must be made of the two arcs of parabola passing through the origin of the phase plane (Fig. 5), and the switching function can be represented by any of the two following expressions:

s = e + ~—-e-lel J,TM

or

.s = \ / - r ^ • |e| • signjfi} + e V ^

(11)

However it is clear that this bang-bang control is very sensitive to model mismatches such as a wrong estimate of the system inertia, friction or spring forces, . . . For instance, if the inertia is underestimated (M < M) there will be overshoots, the transient vanishing in an oscillatory manner, on the contrary, if the inertia is overestimated ( M > M) the system will be driven back and forth across the switching line and it will be sliding towards the origin along this line. This is called a sliding mode and in fact it can be viewed as an equivalent or averaged behaviour on which is superposed an oscillation called chattering [74]. For these reasons it is often preferred to force the system towards the origin of the phase plane in a sliding mode motion by means of a simpler switching function which is a pure linear combination of the position and velocity errors (Fig. 6): s = LO • e + e

(12)

Remark that this control mode could also be presented as a limit case of a P-U control system: r„i = A'„ • (a; • e -|- e), where the gain A',, is increased to infinity but the controller output is bound to stay in the interval {—TM^+Tm). When Ky goes to infinity the transients of the closed-loop linear system are combinations of two exponential modes: one of them vanishes infinitely fast and the other one corresponds to a time-constant approaching oj~^. In practice however due to the bounds on the control variable the first part of the transient is a parabolic trajectory in the phaseplane. This is what really occurs in discontinuous control: at first the system is forced towards the sliding mode motion and the latter corresponds to an exponential decay characterized by a time-constant w " ' . Anyway the motion will ultiinately reach the origin of the phase plane, but then, if the controller is not switched off, in other words if the driving torque can only take the two values —TM and +TM, there will be a limit cycle, that is an oscillation around the origin of the phase-plane. Theoretically both this limit-cycle and the possible chattering motion are oscillations with infinitely high frequency and infinitely small amplitudes. In practice, due to unmodeled dynamics such as small time-constants, dead-times, .. . these oscillations will have finite amplitudes and frequencies. In particular, in case of sampled-data systems the minimum period of the limit-cycle is twice the sampling interval. Such oscillations may be acceptable in other applications of ou-off controllers such as residential beating-cooling systems. In position control of mechanical systems they are generally not acceptable because of problems such as p r e m a t u r e wear of some parts of the system, excitation of undesirable vibrations, . . . A remedy for such a behaviour could be the use of a three-position controller with

462

between the two above "on" positions, a neutral or dead zone where the driving torque is set to zero: Tm = T,n -

0 TM-sign{5)

for for

l^l < e |-s| > e

, , ^ ''

This may eHminate the limit-cycle, in general it does not avoid sliding motions the sliding line s = 0 being now replaced by two parallel lines s = +t and s = —f. In order to eliminate chattering superposed to these sliding motions it is necessary to smooth the transitions from 0 to +rM or —TM'-, this can be achieved by replacing the sign-function in the control law by a sat-function: Tm =

TM

• sat

0

if

.s

<

—f

if

\s\ s

< >

e

;i4)

where: .s iat{-} (

= =

-1 .s f

= +1

if

:i5)

e

but this is nothing else than coming back to a linear control with bounds on the magnitude of the controller output. It should be noted t h a t the above two- or three-position controller is in fact a kind of logical controller: the range of the input variables is divided in two or three classes and the controller o u t p u t is assigned a given value depending on the class the input variable belongs to. Such a partition of the input range can be expressed in qualitative terms, for instance: 5

1,s^1

< < >

<^ e "^ e •^

—f

switching switching switching

variable is negative big variable is very small variable is positive big

In the same way the values of the controller output can be indicated by qualitative labels and assignment of these values can be expressed by literal "If... t h e n . . . e l s e . . . " statements, such as: / / the switching If the switching If the switching

variable is very small, variable is positive, variable is negative,

then apply no driving torque. then apply constant positive driving torque. then apply constant negative driving torqw.

Then the control law is expressed in a pure verbal formulation consisting of a set of conditional statements. This set is nothing else than a rule base. In other words the previous discontinuous or logical controller can be viewed as an elementary qualitative controller and it is a first step towards a pure qualitative control system with arbitrary numbers of classes in the universes of discourse of the input and output variables and a proper number of rules in the controller rule base. Pure qualitative control has a major drawback: due to the abrupt transitions between adjacent classes of the controller inputs there are discontinuities in the controller output whatever the number of classes in the universe of discourse may be; this may induce sliding motions and chattering of the controlled system. For instance, if .s is still the single

463

input variable of the controller and there are A' classes in its universe of discourse the only change with respect to the two- or three-position controller considered above would be that there are now N — 1 parallel shding lines in the phase plane (e, e). In order to eliminate chattering it is therefore necessary to smooth transitions between adjacent classes in the universes of discourse of the input variables. In quantitative control systems this can be achieved by relating the controller output to the inputs by suitable continuous mathematical relationships, which can be linear or not. Such a smoothing can be obtained in a way which may seem more appealing and closer to the human way of thinking by introducing fuzzy logic in the interfaces between the rulebased logical controller and the real world. Then it could be said that fuzzy control is an intermediate stage between simple logical control and more complex continuous control, trying to achieve a trade-ofF between the advantages and drawbacks of these two kinds of control systems: simplicity, robustness, ease of maintenance and low cost are nice features of logical control, but chattering is undesirable in many technical or industrial applications; at the opposite continuous control is more complex, more sensitive to disturbances and to perturbations in the controlled system, more expensive and requiring more skilled technicians for operation and maintenance, but it results in a smooth dynamical behavior of the plant.

4

Fuzzy control of robotic manipulators

Fuzzy logic and fuzzy set theory were introduced by L. Zadeh in the sixties [106]. Less than ten years later applications of these concepts to the control of industrial processes were proposed based on the following arguments: • fuzzy control makes use of qualitative reasoning, thereafter it is able of reproducing the behaviour of a human operator a n d / o r using his expertise, without the limitations of the human behaviour; • fuzzy control requires only a qualitative understanding of the static and dynamic characteristics of the plant to be controlled, using imprecise, vague and possibly ambiguous linguistic representations of the real world; this allows the implementation of a control system without the need of an explicit m a t h e m a t ical model of the plant; • fuzziness in the control system gives it some capability of coping with external disturbances a n d / o r internal perturbations, as well as with uncertainties or variations over time of the plant parameters. T h e pioneering work by Mamdani, van Nauta-Lemke, Zadeh and co-workers in this area is well known [52, 54, 68, 107]. The applications that they considered are mainly stable dynamical plants, such as heat exchangers, engines, cement kilns, water and waste t r e a t m e n t plants, . . . However very soon some authors also investigated the application of fuzzy logic to the control of motions of mechanical systems, although often those are by nature unstable or at least neutrally stable. One of the very first published works in this field is probably the paper by Van Amerongen and co-workers dealing with the design of a fuzzy autopilot for ships [98]. However after this paper it took several years before the publication of other results on the application of fuzzy logic to the control of mechanical systems. The next papers were published in the middle of the eighties and were the premises of an explosion of publications dealing

464

with applications of fuzzy logic in robotics. Those papers can be classed in several categories: • position control of robotic manipulators and more generally of servo systems; see e.g. [37, 58, 63, 65, 72, 77, 83, 94, 99]; in the same area some other papers deal with self-organizing controllers [56, 69, 84, 88, 93]; • force or impedance or hybrid force-position control of robotic manipulators; in some cases there is a two-level hierarchical structure where fuzzy logic is exploited only in a high level of coordination [14, 20, 34, 38, 47, 85, 96, 103]; • control of motions of unmanned vehicles and mobile robots, see e.g. [4, 7, 16, 26, 35, 40, 42, 44, 48, 57, 9 1 , 90, 95];

• path generation for robots, including coordinate transformations and teleop-

eration of manipulators, and operation planning in automated manufacturing systems [1, 6, 18, 46, 73, 87, 97, 108];

• use of fuzzy logic in knowledge-based control systems for robotic applications

and in modeling a n d / o r monitoring of robotic systems [13, 18, 23. 25. 43, 59, 64, 87, 108];

• use of fuzzy logic in vision systems and other sensory devices for robotic applications; see e.g. [17, 39, 62]. In the foUowings only the application of fuzzy logic to the control of servo systems and robotic manipulators will be considered, with the emphasis on position control of a one-DOF servo system. As said in previous sections, the control system of a multi-link manipulator is a collection of joint controllers each of which is dedicated to a single joint and drives it individually.

4.1

Position control of mechanical systems

There is a major difference between the application of fuzzy logic to the control of stable industrial processes and to position control of mechanical systems, because in general the latter are unstable or neutrally stable. In the first case, when a single output of the plant must be controlled, the fuzzy control system can often be compared either to a proportional controller (P) or to an incremental or integral controller (I) or to an incremental P-1 controller. In P-control or in I-control the single input of the controller is the error signal that is the difference between the desired and actual values of the controlled variable, and the controller determines the value of the control variable (P-control) or of an increment of the latter (I-control) according to the current value of this error signal. If the controlled plant is stable both P-control and I-control are allowable, if it is only neutrally stable, in other words if there is a pure integrating process in the plant, I-control would generally lead to an unstable loop. In P-I incremental control the controller determines the control increment on the basis of the current value of the error and also of the error change or the change of the controlled variable, that is the change of the current value with respect to the past value of the signal which is considered. So there are two inputs to the rule base which is the core of the fuzzy controller, one of them stabilizing the closed-loo]) if the

465 controlled process is neutrally stable and the other one providing rejection of disturbances and reset of the controlled variable in steady-state. Then, with a P-controlJer or a P I incremental controller if the controlled plant is neutrally stable, with any of the three controllers if it is stable, the closed-loop system is stable as long as the control action is not too strong, in other words as long as the gain of the equivalent "linear" controller is not too high. The situation is quite different in the position control of most mechanical systems. If there is no viscous friction providing some damping in the controlled system the model of a one-DOF mechanical system is usually a double integrator. Then, as it can he seen by relationships (4) and (5), in order to stabilize the closed-loop and give it appropriate damping, the control law must include a term related to the velocity error or to the actual velocity, both in linear and in discontinuous control systems. This is the same for a fuzzy control system. The simplest fuzzy controller for position control of a mechanical system has two inputs, the first one being naturally the position error and the second one being a velocity signal in order to stabilize the closed-loop: either the actual velocity of the mechanical system or the velocity error or the change in the controlled position or the change in the position error. If the torque or force driving the mechanical system is proportional to the output of the controller the latter can be compared either to a linear P-D controller or to a P-D discontinuous controller. Then assuming no spring force and no external loading force such a control system has no steady-state position error in its normal operating range. However such a control system is unable to cope with disturbances, so that there will a position error due to static loading. The only way to achieve disturbance rejection is the inclusion of an integral action in the control system. This can be obtained via a three-input controller which can be compared to a linear P-I-D controller. Now three terms are necessary in the control system: the first of them is the natural control action, the second one is required for stabilizing the control loop and the third one aims at rejecting disturbances. However, when it is relatively simple to design a two-input rule base or the equivalent decision table, by establishing some correspondence between the decision table and the phase plane of the control system, this becoines more difficult with a three-input rule base. There are some recent results concerning that approach, they are reported in a following section. Another way is the use of a model-based control structure which allows a neat separation of the tracking and disturbance rejection functions of the control system [32, 28].

4.2

Two-term fuzzy control

The control loop of a one-DOF mechanical system the position of which is controlled by a two-term fuzzy controller can be represented by the block-diagram of Fig. 7. T h e control system consists of three parts. The core of the control system is the fuzzy controller itself, that is in this application a two-input-single-output controller which is processing two fuzzy input variables using a rule base and an inference engine. Then, on one side there is a fuzzyfier or Real-to-Fuzzy Interface (RFI) for interpreting d a t a from the real world in the universes of discourses of the fuzzy variables, and on the other side there is a defuzzyfier or Fuzzy-to-Real Interface (FRI) converting the fuzzy controller output into a real control variable. As the controller itself utilizes a rule base and an inference engine it can oidy process discrete-time data. Then the fuzzyfier must include a scaler for mapping the range(s) of the real inpTit variable(s)

466

External Disturbances

Rule base [!

nference engine E„

(Id

•-

' 111

FRI —-< +

E„

RFI

RFI

Kp

K,

<>

Figure 7: Fuzzy control of a one-DOF mechanical system onto the universe(s) of discourse of the fuzzy variable(s), a sampler for sampling data, and possibly also analog to digital converter(s). T h e sequence of these operations depends on the actual implementation of the interface. On the other side the defuzzyfier contains a sample-and-hold device, in general together with a digital to analog converter, and a scaler or amplifier for mapping the universe of discourse of th<^ fuzzy controller output into the range of the real control variable. In position control one of the input variables is the position error, the other one is a velocity signal which can be the actual velocity of the mechanical system or the velocity error or the change in the controlled position or the change in the position error. In the two latter cases the position or error change must be computed in the input interface of the control system. In the two first cases velocity measurement is required in addition to position measurement; moreover the use of the velocity error requires the knowledge of the desired velocity, which is the case for robotic manipulators and some other servo systems where the desired trajectory is programmed in advance and ilesired positions and velocities can be stored in the computer memory. In some papers on fuzzy control of servo systems the universe of discourse of each fuzzy variable may be divided in up to thirteen different classes, in most works this number is limited to seven classes. Sometimes those classes are defined by numbers, but usually the following linguistic labels are assigned to them:

467 Negative small Negative medium Negative big

Zero (ZE) or Around zero (AZ) (NS) Positive small (NM) Positive medium (NB) Positive big

(PS) (PM) (PB)

For simplicity most authors use for the various fuzzy sets membership functions having trapezoidal or very often triangular shapes. However it should be noted that in some works on autonomous drive a n d / o r parking of vehicles the number of fuzzy sets may be different and the shape of the membership functions may be tailored to the type of variable, being different for acceleration, speed and position [50]. Often it is desired to have a fine control for small position errors while control can be coarse for big errors. Some authors deliberately use two different decision tables for that purpose [63]. In many cases it is sufficient to introduce a nonlinear mapping of the range of the real variable onto the universe of discourse of the corresponding fuzzy variable. This can be achieved by means of a piecewise linear function or a sigmoidal function like atan (.). Another way to obtain the same effect is the use of asymmetric triangular membership functions like in Fig. 8 [29].

NB

NM

NSAZPS

PM

PB

Universe of discourse Figure 8: Asymmetric triangular membership functions In a two-input fuzzy control system the control strategy consists of a set of rules or qualitative statements which can be expressed by linguistic or verbal expressions such as: If input N"! is Ej and input N°2 is E2 then apply U control where Ei , E2 and U are linguistic expressions of fuzzy values or labels of fuzzy sets assigned to the input and output variables of the controller. For instance for a position-velocity controller : If position

error is positive big and velocity error is negative then apply positive medium driving torque

small

If there are rn fuzzy sets for the first input and n for the second one the number of such rules may go up to rn * n. For a two-term position controller with seven sets for position and as many for velocity this gives 49 rules. T h e most concise way of representing that rule base is a decision table like Table 1, where there are two entries corresponding to the two input variables of the controller.If one thinks of that table as a qualitative partition of the (e, e) phase-plane of the control loop one sees clearly t h a t there are five "parallel" switching lines in that plane. This analogy between fuzzy logic and multilevel relay control had been pointed by [92]. This set of rules

468

may not give the closed-loop system the best transient performances. The responses in the neighbourhood of the setpoint can be improved by changing some control rules or as in [63] by the use of two lookup tables, one for coarse control the other one for fine control, the controller switching from one lookup table to the other depending whether the error is within given limits or not. Table 1: Decision-Table of a Position-Velocity Fuzzy (,'ontroller

Velocity Error

PB PM PS AZ NS NM MB

AZ PS NS AZ NM NS NB NM NB NB NB NB NB NB NB NM

PM PS AZ NS NM NB NB NS

PB PB PB PM PB PB PS PM PB AZ PS PM NS AZ PS NM NS AZ NB NM NS AZ PS PM

PB PB PB PB PM PS AZ PB

Position Error It is well known that, in fuzzy control on the contrary of pure qualitative control, several control rules can be active at the same time with various degrees of fulfilment and contribute to the final control action to the extent of the latter. So all the active rules must be combined in an appropriate way to produce a resulting fuzzy control action. T h e degree of fulfilment of a given rule is obtained as the minimum or the product of the membership grades of the values of the two input variables which are able to activate the rule. Then the membership function of each rule control action is either clipped to a maximum value equal to its degree of fulfilment or it is simply multiplied by the latter. T h e resulting rule contributions are combined in a final fuzzy control action through the union or the sum of their membership functions, and the control action is defuzzyfied in the defuzzyfication part of the control system. Several defuzzyfication methods can be used, such as the ''nieauof-maxima" method, the "centre-of-gravity" method and the "fuzzy-mean" method. Analysis and experiments [45, 61] show that the first method reduces the "fuzziness" of the control system and may lead to steady-state errors a n d / o r oscillations (limitcycles). So this method is not suitable for fuzzy control. The second method is that which preserves the best the fuzziness of the control system, but its usual continuous form is not suitable in terms of computations for practical applications. The "fuzzymean" method can be viewed as a discrete version of the "centre-of-gravity" method, using discrete representations of the fuzzy sets at the output of the controller and rej)lacing integrals by sums for determining the centre of gravity [45, 80]. Then the defuzzyfied value of the control variable is given by:

u—

(16)

where A^ is the number of tiiscretizatlon levels in the universe of discourse of the controller o u t p u t , u^ is the real value of the controller output attached to the k~ level and 7^ is the value of the membership function of the controller output at UkIf the fuzzy sets at the controller output are crisp and the sum is used for combining the rule contributions the sums in (16) have simply to be extended to all the active

469

rules (that have a non null degree of fulfilment), A' being now the number of active rules and 7^,. the degree of fulfilment of an individual rule. The role of defuzzification methods in fuzzy control is analyzed in [45]. For simplicity and speed of calculations in position control one often chooses the "fuzzy-mean" method with crisp o u t p u t fuzzy sets, summing for all the active rule their contributions weighted by their degrees of fulfilment and dividing the weighted sum by the sum of the degrees of fulfiment. W h a t may be more important for the control system performances is the distribution of the fuzzy values in the universes of discourses of the inputs and the output of the fuzzy controller. Fig. 9 and 10 show the transient responses of the control loop for various distributions of the universes of discourses, which may be different for the inputs and the output of the controller. Fig. 8 is an example of classes that are inequally distributed over the universe of discourse. It can be seen that distributions corresponding to equivalent controller gains which are higher for small errors than for bigger errors result in better performances. Then fuzzy logic appears as a clever way of introducing nonlinearities in the control system in view of better performances. At the opposite if the choice of the universes of discourses for the inputs and the o u t p u t of the controller leads to a fuzzy controller which is equivalent to a linear controller the performances of both are similar as it can be seen from Fig. 9 and 10 . Other ways of introducing nonlinearities and so some intelligence in the fuzzy control system were proposed bv several authors, see e.g. [:?6].

4.3

Three-term fuzzy control

The previoTis theoretical analysis as well as results presented in Fig. 9 and 10 show that position control systems using only d a t a on position and velocity, either with conventional numerical processing or with fuzzy logic, have steady-stale errors when the mechanical system undergoes a static loading. The rejection of this disturbance implies an integral action be introduced in the control system, such that the latter combines three actions, one of them being the natural control action, the second one for stabilizing the control loop and the third one for disturbance rejection. Several implementations of three-term controllers were proposed in the literature. In [58] six different structures of linguistic P-I-D controllers were considered and applied to the model of an industrial process. Authors concluded that for their application controller structures represented at Fig. 11 lead to the best performances. Those structures add up the defuzzified output of an usual P-D fuzzy controller and t h a t of an incremental P-I or I fuzzy controller. Then only rule bases with two inputs or only one single input are to be implemented and the gain of the output stage of the P-F or I controller could be viewed as an additional parameter which can be selected by the designer of the control system. Trial and error procedures or fuzzy translation of the gains obtained by conventional techniques for linear P-I-D controllers are allowable, for instance starting with a P-D controller which gives satisfactory transient performances one could increase progressively the coefficient of the integral action. Application of this work to the control of servo systems is obviously possible. In [86, 88] the controller is an incremental three-term controller determining the increment of the control action on the basis of three inputs: the position error e, the error change c and the rate of chatige in error r. Then ndes with three inputs are required; [88]

470

Figure 9: Tracking responses (ramp-wise reference signal)

Figure 10: Responses to a constant disturbance

1 2 3 4 5

Input space Output space inequally distributed classes equally distributed classes inequally distributed classes inequally distributed classes equally distributed classes equally distributed classes discrete-time linear control equally distributed classes inequally distributed classes

471

presents an application to the position control of a two-link manipulator, using an individual three-term controller for each link; a priori setting of the control rules being difficult, the latter are determined automatically inside a self-organizing controller. e„ , , -J, Fuzzy PD Fuzzy PD ^ Cv T -I-

Fuzzy PI

^

Fuzzy I

Figure 11: Fuzzy PID controller structures (fuzzyfier and defuzzyfier are included in the two blocks]

4.4

Self-organizing fuzzy controllers

A self-organizing fuzzy control system acquires its own control rules through experience. Such a control system is represented in Fig. 12; there are two hierarchical layers: the basic layer is a fuzzy controller with two or three inputs, the other layer is a learning module which generates a n d / o r modifies control rules in order to obtain a predetermined control performance. Modification is achieved in two steps: first, assessment of the control performance, second modification of the control strategy if the performance is not satisfactory. Performance assessment is achieved by means of a performance index which is a measure of the difference between the desired and actual outputs of the controlled system. Performance assessment can be described by linguistic rules similar to the rules of the fuzzy controller itself: / / error is E and change, of error is C then assign value P to the performance

index .

A possible modification of control rules can be based on the assumption that the current value of the performance index is mainly due to the controller o u t p u t n sampling intervals in the past. If the performance index has value zero no modification is needed. On the contrary a non null value of the performance index is a measure of the correction which is required at the system output. Then the linguistic output value of a control rule which was active n sampling intervals before is changed, the new value being equal to tlie previous one plus the current value of the control index. Such a self-organisation of the control rule base was used in two- and three-term fuzzy controllers for position control of three-link manipulators [88, 93]. Moreover in [88] a third layer was introduced in view of automatic self-tuning of the scaling coefficients related to the three inputs of the fuzzy controller (Fig. 13). Self-tuning is based on a small number of tuning rules relating increments of the input gains to variations of some performance ratios (integral squared error, rise-time, % of overshoot). The gains are updated through a cyclic procedure allowing successive refinements of each performance ratio. However an initial guess motivating a stable control loop is necessary to start the tuning procedure. The tuning rules may be defined by analogy with Ziegler-Nichols rules for tuning P-l-D parameters [74] taking into account the peculiarities of the present control structure.

472

R-^Q

Estimate of PI

Learning Module Rule Modifier

Rule base

Scaler

RFI

Inference engine

Fuzzy ('ontroUer Scaler —i

FRI

Basic control loop sctpoint +

Process

<>

Figure 12: Self-organizing fuzzy control system

- R - Q - —^ Estimate of PI

Learning Module Rule Modifier

Estimate of Performances ' Tuning Module Gain Tuner

S Cider

RFI

Rule base

Inference engine

Fuzzy ("ontroller FRI

• Scaler

Basic control looi ) setpoint +

K^ 'CJ^ Figure 13: Self-tuning self-organizing fuzzy control system

473

Self-organizing controllers (SOC) give very good performances for a given task but they may be not so good for other tasks which would very different from the task performed in the learning phase. Moreover it is obvious that learning requires a lot of successive ex])eriments and involves many calculations. Then the use of such controllers is the most efficient when the manipulator has to repeat many times the same task, the latter l)eing known a priori or not.

4.5

Force control of manipulators

Several schemes were proposed in the literature either for force control or for hybrid position/force control or for the control of the mechanical impedance; see e.g. [12, 41, .51, 70, SI, 101]. The major difficulty in force control is that the stiffness or the mechanical impedance which is related to contact forces arising during the manipulator operation is not known or may change; as it may take very high values this leads to stabihty problems for high values of the controller gain or , on the contrary if the controller gain has been set to a low value, to poor dynamic performances resulting in sluggish motions [2, 101]. Because of those difficulties and due to the fact that force control should not be very accurate this problem seems to be a good topic for application of fuzzy logic. Nevertheless it is only recently that this problem has been considered on a fuzzy basis [14, 20, 34, 38, 47, 8.5, 96, 103]. In [38] authors propose a hybrid position/force control using only the normal position control loops of the manipulator and a wrist force/torque sensor. It means that the control system has a cascade structure involving two hierarchical layers: the normal position control loo])s as secondary loops at the lower level and at the higher level force control loops su])plying corrections of the desired positions as it can be seen on Fig. 14. In this control scheme S is a diagonal matrix and each diagonal element being equal to 1 or 0 si)eciiies which degree of freedom in the robot workspace is under force control or position control. Desired positions and displacements are specified and forces are measured in the workspace of the robot. However the motion is obtained by actuation of the joints. Then in the force control loops there must be coordinate transformations from the workspace into the space of joint coordinates and vice-versa. Setting q the vector of joint coordinates and x the vector of linear and angular coordinates in the workspace of the manipulator one has: x^fiq)

(17)

6x^J{q)-bq

(18)

and for small displacements:

where Jiq) is the .Jacobian matrix ,/•' = —-^ . Then, a princijile of virtual works yields dqj the following equalities: T'^ -Sq = F'^ -fix ^ F'^ •J{q)-6q (19) which imply: T = r{q) • F (20) where r and F are the vector of torques and forces respectively on the joints and in the workspace of the manipulator. Therefore the changes in the setpoints of the joint position control loops are: tq^ = J-\q) • (/ - S) • 8x^ -H A'-' • Tj (21) where K = J^{q) • lix " •^(9)1 A'^ is a stiffness matrix in the workspace and TJ is the vector containing the outputs of the force controllers. As it can be seen on Fig. 14 there are

474

several nonlinearities and time-varying gains in the force control loops, moreover there are uncertainties in the stiffness matrix related to the contact forces between the robot and its environment. Then in [38] authors use P-I controllers in the force control lines and they introduce a fuzzy tuning of the gains of these P-I force controllers on the basis of the errors and the error changes in the force control loops. This control scheme vifas tested through simulation on a two-link manipulator in [38].

forces

Robot environment contact forces

setpoints in workspace

Force -|- Position Control System

Joint ControDers

Robot

])ositions in workspace

joint positions -)- velocities

Adaptation Mechanism

desired forces desired positions

PI Controllers

I - S

T/

A'-

J-

setpoints for joint position controllers

Force -t- Position C'ontrol System Figure 14: Hybrid position/force control of manipulators

4.6

Flexible manipulators

As previously said present industrial robots are usually operated at relatively low speeds so that the flexibility of the links and the compliance of the joints can generally be neglected as well as interactions between the motions of the various links. This greatly simplifies the design of the robot control system, which is a simple collection of joint controllers each being dedicated to a single joint and driving it individually. It means that all the effects mentioned above are considered as small disturbances and/or unmodeled dynamics and that the control system is given a sufficient robustness versus all these disturbances and perturbations. However there is a trend towards light-weight robots operating at high speed. In such manipulators flexibiUty of the links and compliance in the joints can lead

475

to vibrations of the links. Then it is necessary to take into account flexibility of the links and compliance in the joints as well as the interactions between different axes in the design of the robot control system. Many papers on the control of manipulators with flexible arms and the suppression of vibratory motions were pubUshed in the last years; see e.g. [5, 15, 22, 66, 100, 104]. Again it is only recently that the use of fuzzy controllers has been considered for that purpose. A study of fuzzy control of a flexible arm is reported in [!)'i]. Authors use a two-term fuzzy controller for position control like in Fig. 15, but they introduce a second two-term fuzzy controller for suppression of structural vibrations on the basis of data supplied by a strain gauge fixed on the arm. Then the outputs of the position controller and the vibration controller are combined in a fuzzy synthesizer which produces a single control variable according to the following decision table (Table 2). Experimental results reported in [53] show that if the strain gauge location is carefully selected this control scheme is very effective for suppressing the effect of the first vibration modes the higher modes vanishing quickly thanks to the internal damping in the arm. It seems that this fuzzy control system may be more robust than conventional optimal control when there are some parameter changes in the controlled system.

Strain + ChaLUge ' Vibration Suppressor Fuzzy Synthesizer

•

Robot

Position Controller Position -f Vel ocit Y Figure 15: Fuzzy control of a flexible arm

Table 2: Decision-Table of a Fuzzy Synthesizer for Vibration Suppression

Vibration Controller Output

PB PM PS AZ NS NM NB

AZ NS NM NB NB NB NB NB

PS PM PB PB PB AZ PS PM PB PB NS AZ PS PM PB NM NS AZ PS PM NB NM NS AZ PS NB NB NM NS AZ NB NB NB NM NS N M N S AZ P S P M Position Controller Output

PB PB PB PB PM PS AZ PB

476

5

Separation of tracking dynamics and disturbance rejection problems

Fuzzy control loops have the same basic properties as conventional quantitative control loops; in particular disturbance rejection in steady-state operation can be achieved if and only if there is a reset action in the controller. If the plant to be controlled is stable it is allowed to iinplenient discrete-time incremental controllers providing at each samphng time an update of the actual control variable based on two inputs: an error signal and the change of this error or that of the controlled variable. This is close to the way of reasoning of a human operator; unfortunately this implies the presence of an accumulator or an integrator in the control line and for closed-loop stability the controller gain should be kept relatively low and the sampling rate rather slow, which forbids fast closed-loop dynamics. Faster dynamics can be achieved only with three term-controllers. Besides, if the controlled system is open-loop unstable or even neutrally stable, which is the case of many mechanical systems and specially of manipulators, three controller inputs are required, one of them for stabilizing the control loop and giving it fast dynamics and another one for rejecting the effect of disturbances. Then the complexity of the closed-loop is increased and it is not so easy to achieve the desired dynamic performances. Moreover an appropriate a-priori weighting of the three control actions in the ride-base of a fuzzy controller may be difficult. However there is another way for solving this problem; it is the use of two control loops, one of them for giving the complete system satisfactory dynamics and stabihty and the other one for rejection of external disturbances and for possible enhanced robustness to internal perturbations and unmodeled dynamics. This is the case in model-based control schemes where there is a neat separation between the two functions: tracking and disturbance rejection of the control system [28]. Various analog or digital control systems based on a model of the controlled plant have been proposed for process control applications since the fifties, see e.g.[3, 24]. Roughly speaking a model-based control scheme is composed by three basic parts: the system that unist be controlled, a controller or compensator driving this system and a model of the controlled system, the difference between the outputs of the controlled system and its model being used for correcting either the output or the setpoint of the previous controller or compensator. Obviously the plant to be controlled cannot be altered, but the control systein itself can be set up independently of the particular implementations of the controller and the model of the process. Therefore as pointed in [29] either the controller or the model or both of them can be implemented in the process computer using fuzzy and qualitative control concepts. Hybrid implementation including fuzzy or qualitative implementation of some parts of the control system and numerical or quantitative implementation of others is also possible. Besides different sampling rates may be used in different parts of the control system, for instance in the basic controller and in the model, which results in a multi-rate control system. The analogy of such a control scheme with the human behaviour is straightforward. It can be reasonably thought that the control strategy of a human operator is based on a qualitative understanding of the process that he has to control: in other words the oi)erator would have in his mind a qualitative model of the process response, the latter being either a static model (gain ) or even a dynamic model ( the process i.<; stable or not, it.s traii.sient response is oscillatory or damped, the operator has a rough estimate of the ])rocess settling time, . . . ) . Then a human operator would correct the manipulated control variable according to the deviation of the actual process output from the output which had been forecast by this internal model. Furthermore the basic control action taken by the operator and the correction of this action can be performed at different time scales. Therefore the

477

introduction of fuzzy or qualitative models in control systems is a natural extension of the use of fuzzy concepts in control. There may be different model based control schemes, depending on the way the control system uses the "return-difference" signal (difference between output signals of the actual controlled process and its model; note that the output of the actual process is not necessarily the controlled output variable, it can be an auxiliary output variable, but obviously the latter must be measurable). First, this signal can be used for correcting the setpoint of the basic controller like in Fig. 16, the control input being the same for the controlled process and its model. This control scheme is known as "Internal Model (Control" (I.M.('.) and is probably the oldest and the best known model-based control system; see e.g. [71]. In two other schemes, the return signal is used for correcting only either the control input of the model or that of the controlled process. In the first of these two schemes the return signal is used for forcing the state of the model to track that of the actual system, in fact it is nothing else than the well known Luenberger observer [67]. The second of these two schemes (Fig. 17) is the dual of the first one and as such, it has the same nice properties as shown in [32, 60]. It can be said that in this control scheme the return signal aims to force the output of the actual system to track that of the model and the effect of unmodelled disturbances acting upon the system is counteracted by the effect of the return signal. The latter scheme is nothing else than "Model Reference Control" or "Model Following Control" [60]. disturbance

1

reference

Process

process on tput

cs)

tn V' . Model

Correction Mechanism

model out put

return signal

Figure 16: Internal model control As shown in [32] all these control schemes have the same basic properties: first, the nominal tracking dynamics is the same whether the return signal is used or not in the control system; second, the regulation dynamics can be tailored to the designer's objectives, thanks to a suitable filter processing the return signal; third, the three schemes are equally robust. The first pro])erty is obvious, as in case of exact model match and no disturbance the return signal is null and therefore it cannot change the tracking dynamics of the controlled system. As a consequence of that property control schemes of Fig. 16 and 17 can be used as such if and only if the system to be controlled is open-loop stable. If not there must bo another feedback loop for stabilizing the system and giving it the desired dynamics. This loop may be either a secondary loop inside the model-based control scheme or an external loop around it. In the first case the model must be that of the resulting stable closed-loop

478 system. On the contrary, in the second case, the model is that of the system which must be controlled and it allows a correction of the control input to this system. Under some general conditions this correction can be seen as an estimate of an equivalent disturbance signal acting upon the system input. Then the latter model-based control scheme is also called a "disturbance observer". Mechanical systems are generally unstable or neutrally stable. Then several model-based control schemes are allowable for position control. The case of a one-DOF servo system is considered in the followings. disturbance

Process

reference

C'orrection Mechanism

Model

process output

return

signal y ^

model output

Figure 17; Model following control

5.1

Position control with disturbance observer

In this control scheme there is inside the main control loop stabiUzing the process and/or giving it the desired dynamics, a disturbance observer using a secondary feedback loop (Fig. 18); the main loop is a two term (position + velocity or P-D) control loop, the secondary loop may be a velocity control loop only. The model of an ideal servo system being a double integrator this control scheme appears as cascading a P I compensator with the main I ' D feedback controller and implementing a secondary feedback loop inside the main loop, in such a way that the dominant pole of this secondary loop is aiitomatically cancelled by the zero of the P-I compensator. Then the use of this disturbance observer results in a complete rejection of steady-state disturbances; simulation and experimental results showed the effectiveness of this control scheme for the control of servo systems and robotic manipulators, whether the main controller and/or the model were implemented as numerical or fuzzy systems [9, 33, 30, 29, 31, 82]. In [30, 31, 82] the main controller and the disturbance observer were conventional quantitative systems either in continuous- or in discrete-time. In [9, 33, 29] the main controller was a two-term fuzzy controller using fuzzy values of position and velocity errors, and inside the main control loop there was a disturbance observer based on the model relating the velocity to the driving torque: with the above assumptions this model is a simple integrator. Several implementations of this model were tested: numerical, qualitative or hybrid (combination of the two previous ones). All these different implementations of the main controller and the model gave similar results [8]. The effect of the model-based control on disturbance rejection is clearly seen in Fig. 19.

479

disturbance

reference

process outputs

Process

^

Correction Mechanism

Main (Controller

Model

return/eturuAN sign ienalV. f +

model output

Figure 18: Control with disturbance observer inside the main control loop

0.5

0.45 0.4

-

fl

D = 0.5

p O 0-351 s i 0.3-t

D = 0.25

0.1

\j\

X^

n E 0-2 ^ r r o 0.15 -• r

D = 0.125

-

\

ill

i1 \

1

ij

0.05 (;/

'1

'r '•''•

/i^l'"'--,—.

|/A 10

15

20

1 25

30

. ""••-35

40

45

Time [s] Figure 19: Responses to constant disturbances: D = 0.125, 0.25, 0.5 Full lines: Dashed lines:

fuzzy PI) control fuzzy PD control + model-based control

480

5.2

Model reference control of positioning systems

Another way to cope with disturbances in neutrally stable or unstable systems is to equip them with a stabilizing feedback loop and to use internal model control (Fig. 16) or a model following control (Fig. 17) for the resulting stable closed-loop system. In position control of a one-DOF servo system stabilization and the required dynamics are obtained via an auxiliary position + velocity feedback loop, using either a numerical or a fuzzy controller. Outside this stabilizing loop there is a qualitative model of the desired response which can be used either in the internal model control scheme of Fig. 20 [10, 11, 27, 29] or in model following control as in Fig. 21 [29]. The model can be viewed as a one-step ahead predictor disturbance

o

Stabilizing CJontroUer

process output

Process

6= Model

model output

Correction Mechanism

return signal

I-'igurc 20: Internal mode! control of a process stabilized by a secondary feedback loop disturbance

Stabilizing Controller

reference

Process

Correction Mechanism

process output

return A signal " v .

model output Figure 21: Model following control of a process stabilized by a secondary fecdb

loop

481 which, at each sampling time, predicts the next value that the actual system output should reach given the current data, and the difference between predicted and measured values is used for correcting the control variable at the next sampling time. Due to the robustness of model-based control schemes modeling must not be accurate: an approximate model representing roughly the static gain and the settling-time of the controlled system should be satisfactory in most cases. If the sampling interval for the model-based control scheme is at least equal to the settling-time of the closed-loop controlled system the simplest model is even a one-step ahead shift operator, (it can be seen as a "wait and see" strategy leading to small updates of the control variable). This model can be embedded in any of the two control schemes of Fig. 16 and 17. It was shown in [11, 29] that the disturbance rejection capabibties are the same in the two cases (Fig. 22), but obviously the errors when tracking time-varying signals are different; in particular for a ramp-wise varying desired position the internal model control scheme leads to a constant error which is nothing else than the so-called "velocity error" defined in textbooks on control systems (see e.g. [74]); this error could decreased or even cancelled at the price of an increasing complexity of the model. Another model which can be used in the control schemes of Fig. 16 and 17 is a first-order predictor with a sampling interval approximately equal to half the rise-time of the closedloop controlled system [11, 29]. Obviously decreasing the sampling interval allows faster reactions of the model-based control system as it can be seen on Fig. 22 (curve in dotted Une).

0.5 0.4

D = 0.25

D = 0.125

-0.2 -0.3 0

10

20

30

40

50

60

70

Time [s] Figure 22: Responses to constant disturbances: D — 0.125, 0.25, 0.5 Fxdl lines: Dashed lines: Dotted lines:

model following control internal model control with one-step ahead shift operator internal model control with first-order predictor

482

5.3

Implementation of qualitative or fuzzy models

A complete fuzzy implementation of the above control systems requires qualitative adders and subtracters and above all either a qualitative or fuzzy implementation of the integrator in the disturbance observer of Fig. 18 or that of the model used in the control schemes of Fig. 20 and 21. The rule-ba.se of a qualitative adder can be represented by the decision table given as Table 3. Obviously the same decision table is valid for a qualitative subtracter provided that each of the values of the subtracted variable is replaced by its opposite value. Table ',i: Rule-Base of a Fuzzy .^dder

Input 2

PB PM PS AZ NS NM NB

AZ NS NM NB NB NB NB NB

PS AZ NS NM NB NB NB NM

PM FS AZ NS NM NB NB NS

PB PM PS AZ NS NM NB AZ Input 1

PB PB PM PS AZ NS NM

Ps

PB PB PB PM PS AZ NS PM

PB PB PB PB PM PS AZ PB

The discrete-time representation of an integrator is an accumulator the output of which is u])dated at each saTUpling time by combining the last values of the output and input data: (22)

Vu = y + M

where u and y are respectively the input and the output variables and subscript u means an u])dated value. (Uearly the accumulator is nothing else than an adder with a regenerative feedback loop as shown in Fig. 23, and its qualitative implementation can be represented by the same rule-base as in Table 3 just replacing one of the inputs by the last output (Table 4). 1

^

Backward Time shift ^

F A^

¥•

Figure 23: Fuzzy accumulator (F.a. = fuzzy adder)

Table 4: Rule-Base of a Fuzzy Accumulator

Last Qniput

PB PM PS Z NS NM NB

AZ NS NM NB NB NB NB NB

PS AZ NS NM NB NB NB NM

PM PS AZ NS NM NB NB NS

PB PM PS AZ NS NM NB AZ Input

PB PB PM PS A'Z NS NM PS

PB PB PB PM PS AZ NS PM

PB PB PB PB PM Pb AZ PB

483

However pure fuzzy implementation of the neutrally stable closed-loop subsystem shown in Fig. 23 would lead to an explosion of fuzziness, i.e. the support set of the output variable would increase at each sampling time. Explosion of fuzziness is due to the type of fuzzy rules "If X is A then Y is fl" used in the implementation, namely possibility riiles. Semantic behind possibility rules is : the more X is A , the more possible Y is B. If several rules are triggered, i.e. several control actions are possible, the union of these actions is possible. More precisely in the case of the fuzzy accumulator, if the input and the last output of the accumulator have respectively two and q simultaneous values the use of a decision table like Table 4 may lead to ^ -|- 1 simultaneous values of the updated output. Then the number of possible simultaneous values of the output could increase at each sampling time! Obviously this number has an upper bound which is the number of classes in the universe of discourse of the output variable, but it means that ultimately the accumulator output would have simultaneously all the allowable qualitative values! Therefore the information would be scattered over all the universe of discourse, which means a complete loss of information. This is what has been called "explosion of fuzziness" [9]. Such situations could be avoided by using another type of rule like certainty or gradual rules [21]. Implementation of the operations related to these types of rules being harder, it was chosen to roughly limit the fuzziness of the output: in the feedback loop of the accumulator the qualitative variable is defuzzyfied at first, refuzzyfied immediately after and then fed back to the adder (Fig. 24). Then the quaUtative values of the accumulator output which are fed back to the adder keep some membership grades, and this prevents the explosion of fuzziness.

Backward Time shift

Fuzzyfier

Vu

Defuzzyfier Vu

Figure 24: Hybrid fuzzy accumulator (F.a. = ftizzy adder)

Table 5: Rule-Base of a First-Order Fuzzy Predictor

Last Output

PB PM PS AZ NS NM NB

PS AZ NS NS NM NB NB NB

PM PS AZ NS NM NM NM NM

PM PS AZ NS NS NS NM NS

PM PM PS PS AZ PS AZ PS AZ AZ NS NS NM NM AZ PS Input

PM PM PM PS AZ NS NM PM

PB PB PM PS PS AZ NS PtJ

The risk of explosion of fuzziness is weaker in a first-order predictor. In a numerical implementation of the latter the updated value of the output would be a convex combination of the last values of the input and output variables: 2/u = A-t/ + (l - A) •«

(23)

484 which results in coniplenientary attenuations of the input and feedback variables. If A has a value around 1/2, in other words for values of the predictor time-constant approximately equal to the sampling interval, the rule-base corresponding to qualitative implementation of the predictor can be represented by Table 5. Here the attenuation of the feedback variable lowers the regenerative effect and therefore the risk of explosion of fuzziness.

6

Conclusions

The numerous works which have been quoted in this chapter prove the effectiveness of fuzzy control for position and/or velocity control of some mechanical systems such as servomechanisms, manipulators, robots, mobile robots, unmanned vehicles, . . . Most often the fuzzy controllers used in those applications are position -|- velocity or P-D control systems and their rule bases result from a straightforward qualitative interpretation of multilevel relay control rules. Then the question is: as P-D hnear controllers are so well known and so simi)le is it worthwhile to introduce fuzzy controllers in such appUcationsV There is no unique answer to that question. For one-DOF servo systems with known parameters linear controllers and fuzzy controllers with uniform distributions of the values in the universes of discourses perform equally well. However even in that case modifications of those distributions allow the introduction of nonlinearities in the control system, which may lead to better performances. Moreover if there are uncertainties or variations over time of some parameters of the controlled servo system or if there are some nonUnear feedback terms such as friction or gravitational forces a fuzzy control loop may perform better than a numerical control loop, due to the fact that fuzziness implies by itself some robustness against the influence of such pertnrbations. This is still more true for perturbations due to unmodeled dynamics such as compUance in a mechanical transmission or flexibility of an arm. In the latter case the work reported in [53] seems to be very promising as the control is based on two separate two-term fuzzy controllers, one for usual position-|-velocity control and the other one for vibration suppression. This idea of separating different tasks of the control system is basically the same as that which can be found in the works on separation of tracking dynamics and disturbance rejection in [9, 10, 11, 33, 27, 29, 28]. After all those ideas are nothing else than apphcations of the ancient separation principle: ''Divide ut iniperaa" which should be the first of the basic C'omiuandments for any control engineer who is facing tlie problem of designing a control system for a complex plant. This is the same idea which is the basis of most control systems for multi-DOF manipulators where each joint is actuated by its own controller the interactions between the motions of different jiarts being considered as external disturbances. In that case also fuzziness in the control system may imi)rove the overall performance. Eventually there is another case where fuzzy controllers are more attractive than conventional numerical controllers, it is the case where the control objectives and/or the control strategy are only defined in loose terms; examples are not uncommon: driving and/or parking a vehicle, path planning and control of unmanned vehicles or mobile robots, force control, collision avoidance and/or cooperative tasks for robotic manipulators. Sucti a])])lications were considered in several papers, see e.g. [47, 78, 102] and it is likely that further developments of fuzzy controllers are to be expected in those areas. In most of the works presented in the previous paragraph fuzzy control was applied at the lower level of acttiation of a mechanical system that is position and/or velocity control. On the contrary, in many of the a]>plications mentioned here fuzzy control is often applied at a higher level in a liierarchical control system. In some apphcations also fuzzy concepts are used for scheduling or tuning the parameters of numerical controllers or for combining the outputs of several parallel controllers.

485 For instance in [89] a robot with a compliant joint is controlled by four controllers, each of them having been calculated for a given area of the robot workspace, and the final control variable is a fuzzy convex combination of the outputs of those four controllers. In [38] the setpoints of some of the conventional position controllers of a manipulator are supphed by P I force controllers the gains of which are tuned by a kind of fuzzy adaptation mechanism. Sometimes also the controUer includes a numerical controller and a fuzzy controller acting in parallel. This basic idea of such a hybrid control scheme is that numerical control and/or compensation should be used for the known part of the controlled system and fuzzy control introduced in order to cope with the additional unknown or uncertain parts. Such a hybrid controller was used in [49] for position control of a servo system. In some sense all these schemes can be related to the fuzzy supervision of control systems, see e.g. [75]; this is also an area of active research.

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[101] D. Whitney. Historical perspective and state of the art in robot force control. In IEEE Conference on Robotics and Automation, pages 262-268, Malaga, 1985. [102] Hyun Woong Keun, II Hong Suh, and Kyong Gi Kim. Intelligent control of redundant manipulators in an environment with obstacle. In IROS' 91 International Workshop on Intelligent Robots and Systems, jiages 161-166, 1991. [103] Y. Xu, R.P. Paul, and H.Y. Shun. Fuzzy control of robot and comphance wrist systems. In IEEE Indu.ftry Applications Society Annual Meeting, pages 1431 1437, 1991. [104] W. Yim, J. Zuang, and S. Singh. Experimental dual-mode control of aflexible robotic arm. International Journal of Robotics and Autoination, 10:135-145, 1992. [105] T. Yoshikawa. Foundations of Robotics: Analysis and Control. Cambridge, Mass., 1990. [106] L.A. Zadeh. Fuzzy sets. Information

The MIT Press,

and Control, 8:338-353, 1965.

[107] L.A. Zadeh. A rationale for fuzzy control. Trans.A.S.M.E.,

ser.G, 94:3-4, 1974.

[108] H.J. Zimmermann. Approximate reasoning in manufacturing. In A. Kusiak, editor. Intelligent Design and Manufacturing, pages 701-722. Wiley-Interscience, New York, 1992.

CHAPTER 18 FUZZY CONTROL FOR ROBOT MANIPULATORS WITH ARTIFICIAL RUBBER MUSCLES Keigo Watanabe^*), Sangho Jin(*) and Spyros G. Tzafestas(**) (*)

(**)

Department of Mechanical Engineering, Faculty of Science and Engineering Saga University Honjomachi-1, Saga 840, Japan Intelligent Robotics and Control Unit Department of Electrical and Computer Engineering National Technical University of Athens Zografou, Athens 15773, Greece

1. I N T R O D U C T I O N W h e n controlling a robot manipulator by applying the well-known c o m p u t e d torque control law, we usually need the mathematically accurate model. However, it is not necessarily easy t o make a rigorous model, because the dynamic modeling of the robot manipulator essentially include some uncertainties such as system par a m e t e r s , disturbance inputs amd nonlinear elements. T h u s we can be faced with t h e problem of designing a robust controller t h a t achieves a stable control, even t h o u g h we use inaccurate modeling information. Some different approaches to the problem of robust control design for uncertain systems have been proposed for a case when the botmds on the uncertainties are known (Hui and Zak, 1992); two major approaches to the deterministic control of uncertain systems are the deterministic control using Lyapunov functions (Corless and Leitmann, 1981: Corless,1989) a n d the variable structure control (or sliding m o d e control) m e t h o d s (Utkin, 1977; DeCarlo et al., 1989). T h e approaches without using the b o u n d information on the uncertainties can also be found in Chen (1990) or Imura et al. (1991). A fuzzy control is a t t r a c t e d as a practical control strategy for controlling a robot manipulator, because we need no m a t h e m a t i c a l models a n d can easily apply it to nonlinear systems as well as linear systems. It should be noted, however, t h a t , when designing a fuzzy logic controller, we must t u n e some controller p a r a m e t e r s such as scalers (or gains) for the input d a t a and the o u t p u t d a t a of the fuzzy logic controller. For this problem, some self-organizing fuzzy controllers ( S O F C s ) have been already examined by several authors (Procyk and M a m d a n i , 1979; Yamazaki a n d Sugeno, 1984; Daley and Gill, 1986; Linkens and Hasnain, 1991; Tanji and Kinoshita, 1987; M a e d a and Murakami, 1988). On the other hand, some (iterative or repeated) learning-type fuzzy controllers (LFCs) t h a t incorporate a neural network have been also reported recently ( W a t a n a b e and Ichihashi, 1990; Hayashi et al. 1990; Horikawa, et al, 1991; W a t a n a b e et al, 1992; W a t a n a b e and Tang, 1992). In this study, we apply a fuzzy logic controller for making a robust control system of a robot manipulator with rubber muscles. We first consider a fuzzy logic 493 S.G. Tzafesias and A.N. Venetsanopoulos (eds.) Fuzzy Reasoning in Information, Decision and Control Systems, 493-510. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.

494 controller as a robust compensator in the problem of tracking control of the robot m a n i p u l a t o r , in which it is assumed t h a t the computed torque control m e t h o d has already been included. We next apply an iterative L F C in which several elemental fuzzy controllers with incompletely t u n e d p a r a m e t e r s are processed in parallel and t h e degree of the usage of each inferred consequent is determined by using a linear neural network, which is called a multiple fuzzy controller. Consequently, when there are modeling uncertainties such as the m o m e n t s of inertia a n d / o r there exist u n t u n e d feedback gains, we will show t h a t two fuzzy control m e t h o d s presented here are effective for controlling a robot manipulator with r u b b e r muscles t h r o u g h some simulations. 2. C O N T R O L

OBJECT

T h e r u b b e r muscles considered are RUB-825 and RUB-420 which are m a d e by Bridgestone. T h e personal computer, as a digital controller, is a PC-9801 FA, whose cpu is 486sx (16 MHz) with an overdrive processor. T h e m a n i p u l a t o r is restricted so as to move only on the horizontal plane, so that the gravitational effect is neglected. The m a n i p u l a t e d voltage from the computer is first transformed into an analogue quantity and further t r a n s m i t t e d into a V / A converter. By supplying t h e m a n i p u l a t e d electric current signal (4 ~ 20 [mA]) to the servo-valve, the electromagnetic proportional valve can control the inlet air flow a n d therefore the inner pressure (0 ~ 6 [Kgf/cm'^]) of the rubber muscle can be changed. E a c h joint angle of the m a n i p u l a t o r is measured by a rotary-encoder (2000 P / R ) and the measured signal is fed back, through a pulse counter, to the computer. Figure 1 shows t h e block diagram of t h e total control system and Fig.2 depicts the action principle of t h e r u b b e r muscle.

[Compressor J

Regulator

' Personal D/A computer -»- converter PC-9801

V/A converter

—»-

Servo valve

—^- Manipulator

1

-

Encorder

Fig.l Control system for a manipulator with r u b b e r muscles

495

Reducing pressure Air

Air - ^ Adding pressure Fig.2 Action principle of a rubber muscle

3. C O M P U T E D T O R Q U E C O N T R O L L E R As a mathematical model of the rubber muscle manipulator, consider the usual SCARA-type robot manipulator model described by M(f)0 + V{0, ff) = T

(1)

where 0 e TZ" is the joint angle vector,^ T e W is the driving input, M(ff) £ K"''^" is the positive inertia matrix and V(0,0) <E K" is a iionHnear term which represents the Coriolis force, centrifugal force and gravitation, etc. It is assumed that we set the desired reference joint angle 64, angular velocity 0i and angular acceleration 64. H we have the output joint angle 9, the angular velocity 0, and the model matrices M(l?) and V(ff, 0), then the computed torque can be constructed by

T^M{0)0*

+V(0,6)

r - 0,1 + K„e + Kpe

(2) (3)

where Ji„ £ ^ " X " jg the different ial gain of diagonal matrix, Jlp € K"^" is the proportional gain of diagonal matrix, and e(t) = [ei(*),.... Cn.]"^ denotes the tracking error vector defined by

e(t)^0dt)^m

(4)

If matrices M{ff) and V(9, 0) are completely identified as actual matrices M(6) and V{6, 0), then the equation of the error vector satisfies tlie following linear equation: B{t) + K„6{t) + Kpe{t)

(5)

496

Fuzzy compensator

Torque computation II

Robot

e e

a

Fig.3 C o m p u t e d torque control system with a fuzzy compensation

It is readily found t h a t , when K„,Kp stable. However, this relation m a y be a n d / o r V{d,'B) ^ V ( ^ , ^ ) . Furthermore, are not optimally t u n e d , we will not be of error response, rather we will have A/(^) = M{6) and ¥{9,6) = V{eJ).

> 0, the equation (5) is asymptotically destroyed for a case when M{d) ^ M{6) for a case when the feedback gains K^,Kj, able to obtain a relatively fast convergence a oscillatory error response, even though

4. D E S I G N O F A F U Z Z Y L O G I C C O N T R O L L E R T h e purpose of this section is to design a robust control system, as shown in Fig.3, by applying the fuzzy logic compensation, under the assumption t h a t a comp u t e d torque control law is simultaneously appHed t o the n-link robot manipulator. 4.1 Dependent Reasoning Method T h e fuzzy logic controller as a compensator simphfied fuzzy reasoning (Mizumoto, 1989), in of M a m d a n i ( M a m d a n i , 1974, 1976), b u t with a p a r t . Following this reasoning, we can write any

used here is described bj' using a which the basic reasoning is that constant value for the consequent i-th control rule as

R, : If Cj = Ail and ... and e„ = A,„ and ei = Ai(n+i) a n d ... a n d e„ = then Ml = 5,1 and ... and w„ = B,n

A^^n) (6)

for 2n-input variables ( e i , . . . , e„, e j , . . . , e„) and n-output variables ( u j , . . . , Un). Here. R, is the i-th control rule, A,j is the fuzzy set of antecedent part corresponding to

497 the j-th input variable in the i-th control rule, and B,j is the constant value corresponding to the j-th o u t p u t variable in the i-th control rule. Using 2n confidences fiAniei),..., fj.An„^{en), fJ-An^^i^iei),..; tJ.An2„)(^n), we have t h e confidence of the antecedent part in the i-th control rule, hi, as follows: h, = /i^,i(ei)A/J^,.,(e2)...A/iA,„(e„)A/i^.,„^;j(ei)A/iA,(„+j,(e2).-A^A,(,,,,(en)

(7)

where A denotes the minimum operator. T h e n , the j - t h inferred consequent is given by

« ; - ^ ^ '

, = i,...,„

(8)

where r denotes the total number of control rules; generally r = f^" if the n u m b e r of labels is i. In order to transform the input variables to the values on the support set, for example, on [—6, 6], we need the scalers for the i n p u t - o u t p u t d a t a of the fuzzy reasoning. Here, we define t h a t GIN(i),? = l , . . . , 2 n are the input scalers for the input d a t a e i , . . . , e„, ii,..., e„ and G O U ( i ) , i = 1,..., n are the o u t p u t scalers for the inferred consequents u*, ...,u*. 4.2 Independent Reasoning Method Generally, the trajectory tracking control problem for the multi-link manipulator is of multi-input multi-output. As seen from the above statement, if we apply the usual (dependent) fuzzy reasoning method, then the number of control rules increases exponentially as the numbers of fuzzy label and link increase. Therefore, if we use t h e dependent reasoning m e t h o d , then a successful fuzzy controller can not be necessary designed. One solution to this problem is to press the properties of a multi-link manipulator into the I F - T H E N forms consisting of some eff"ective rules bj- using the so-called fuzzy modeling technique. T h e other solution to the problem is to use the so-called independent joint control m e t h o d which is adopted for many industrial robots. From the fact t h a t the equation of motion for the multi-link m a n i p u l a t o r is not decoupled each other, this m e t h o d is considered to be an approximate technique b u t has a merit that n single i n p u t - o u t p u t control system can be independently designed for the n-Iink manipulator. Following this approach, we can design the fuzzycontrol rule such t h a t the trajectory tracking error of the j-th link and its derivative are the inputs to the reasoning, and the corresponding consequent is an input torque to the j ' - t h link, if the j - t h reference joint angle and its angular velocity are specified. Consequently, we can n times apply the same fuzzy control rule to the n-link case, but with different i n p u t - o u t p u t scalers. This concept is depicted in Fig. 4.

498

GIN(l) GOU(l)

Calculation of minimum grades

Ae

Defuzzification

GIN(3) GIN(2) Calculation of minimum grades

Ae

^ GIN(4)

^ \

G0U(2) G( Defuzzi- U ^ fication

2f

h^

Fig.4 Block diagram of two independent fuzzy controllers

4.3 Adaptive Input Scaling T h e initial constant input scalers can be easily determined from the m a x i m u m value of the d a t a which are obtained through the control experiments using only the computed torque control m e t h o d . However, we sometimes fail to transform the input d a t a into the predetermined support set, because of the addition of the fuzzy compensator to the control system. To overcome the above problem, we propose here a simple adaptive m e t h o d for scaling the input d a t a e i , . . . , €„ described by 0.91

if |GIN(i)oM X e,| > I

GIN(One

(9)

GIN(i)oia otherwise This means t h a t if the scaled error of the joint angle is scaled out from the support set [ ~ i , X], then the input d a t a is rescaled to fall on the 90% range of the support set. Similarly, the scalers GIN(i + n ) , . . . , GIN(2n) for the input d a t a (i.e., the error of the joint angular velocity) e j , . . . , e „ can also be adaptively determined by the similar condition. 5. Multiple Fuzzy Controller Consider the following nonlinear plant with n-input and ?i-output vectors

x{t) =

f(x{t),u{t),t)

(10)

499

9{t) = g{xiilt)

(11)

where x(t) is the 2n-dimensional vector, u{t) is the n-dimensional input vector, u{t) — [ui{t), ...,«„(<)], /(•) is a 2n-dimensional nonhnear vector-valued function, g{-) is an n-dimensional nonlinear vector-valued output function and 6{t) is the n-dimensional plant output vector 6{t) = [Oi(t),..., 6n{t)] • 5.1 Construction Method When defining the reference vector as Oi(t), we iise the output deviation eit) ^ 64t) - 9{t)

(12)

and the corresponding rate e(i) (or increment Ae{kT)) as the inputs to the antecedent part of the fuzzy reasoning, where A = 1 — z~^, z~^ is the one-step delay operator, k is a positive integer and T is a sampling period. We then drive M elemental fuzzy controllers in parallel with incompletely tuned controller parameters, e.g., the input and output scalers GIN(?), i = 1, ...,2n and GOU(?),; = 1,..., n. Note here that we further allow a situation where each elemental fuzzy controller has a different number of fuzzy labels, a different width of membership functions, and a different number of control rules, etc. The control input to the plant can then be constructed by M

u{t) =

'^w,Ui{t)

(13)

1=1

where Ui(t) — [u\(t),...,u'^{t)] ,i = 1,...,M denotes the scaled inferred consequent generated from the i-th elemental fuzzy controller and w, is a weighting factor. 5.2 Learning Method Among existing several learning architectures (Psaltis et al., 1988), we will adopt here the so-called specialized learning architecture. That is, for the learning of the connection weight u;;, we consider the cost function consisting of the squared norm of the output deviation,

J{k)^\Y.\\e{kTW

(14)

and construct the learning algorithm based on minimizing this function. Here, N denotes the total number of samplings. Figure 5 shows the block diagram of a multiple fuzzy control system. From equations (12)-(14), the gradient of J with respect to w, becomes dJ{k) dJ{k) de{kT)du{kT) dwi " de{kT) du(kT) dw,

*:=0

^

'

500 Learning mechanism +

e

Fuzzy controller 1

[^Le

«i

Plant H

Fuzzy controUer M

Uu

Linear NN

Fig.5 Block diagram of a multiple fuzzy controller

Therefore, using the delta rule (Rumelhart et al., 1986) gives the following u p d a t e equation for w,: N

de{kT)

o,{i +1) = «,.(/) + , Y. ^^(^^)^;7Pf("'(^-^) dv{kT)

;i6)

fc=0

where / denotes the l-ih. u p d a t e time and r; is a small positive constant. Since in the application of a fuzzy control the plant structure or p a r a m e t e r s may be unknown, we will evaluate the Jacobian 89 jdu in equation (16) as de,{kT)

AOiikT)

dujikT)

Auj{kT)

l,...,m,j = l,...,p

(IT)

where Auj{-) and A^i(-) are generated from the input a n d o u t p u t data. If the plant is originally a discrete-time system with no any time-delay of the input, then we must evaluate d6i[kT)/duj[(k — 1)T] instead of (17). Note however t h a t if the plant s t r u c t u r e and parameters are completely known, then we can obtain the Jacobian in an accurate form. Note furthermore t h a t we can readily apply the s t a n d a r d linear identification algorithm, for example, by the K a l m a n filter ( W a t a n a b e f.t al., 1991) t o estimate toi, because equation (13) is linear with respect to iv,.

6. E Q U A T I O N O F M O T I O N F O R T H E

MANIPULATOR

We consider a two-link manipulator model as shown in Fig.6. T h e equation of motion is assumed to be given by (1), but with M n =Ii+l,+

m,l]^

+ m2illi

+ ll^ + 2/„Wj2Cos(^2))

501

joint 1 Fig.6 Two-link m a n i p u l a t o r model

Mi2

= M 2 1 = I2+

m 2 ( / j 2 + lnllg2COs{e2)),

V, = -m2lnilg2sm(e2)i29i

+ 62)82,

M22

= h

+ "^2/^2

V2 = m2/„W,2sin(6l2)^f

where M = [Mjk],V = column{y^}(i = 1,2; it =- 1,2), 6 = [6»i 02^, r = [''"1 ^•2]'^- Here, 6^ is the j - t h joint angle, r, is the driving torque a r o u n d the j ' - t h joint, Ij is the moment of inertia for the j-th. link, rrij is the mass of t h e j - t h link, l„j is the length of the j-th link, Igj is the distance from the joint j to the center of gravity of the j - t h link. T h e measured real parameters of the m a n i p u l a t o r are given by Table 1. 7. S I M U L A T I O N

EXAMPLES

To simulate the motion for the above r u b b e r muscle manipulator, we use the 4th-order Runge-Kutta-Gill m e t h o d . It is assumed t h a t the control sampling period is T = 25 [ms], the step width of t h e integration is 0.5 [ms], and the initial states are 01 = 0.7304 [rad], 02 = 0.4858 [rad], ^i = ^ 2 = 0 . Further, assume t h a t the reference trajectories

Xd{t)

A

x{t)

y{t)]

in the work-coordinate systems are given by

x{t) = 25cos(7ri/5) + 300

(18)

502

Table 1 Real parameters of a r u b b e r muscle m a n i p u l a t o r

Mass m , (kg)

Inertia li (kgm^) nominal

Link 1 Link 2

NB

1.39 0.65

actual

0.02734 0.03281 0.00117 0.00189

NM

Length Length of c. /„,(m) Igi (m)

0.385 0.110

ZO

0.160 0.057

PM

PB

0

-12 Fig.7 Membership functions for 5 labels

(19)

y(t) = 25sin(7r
which are the circular trajectory with 25 [mm] radius, and the corresponding cycle is 10 [s]. From solving the inverse kinematics problem, the reference angles Oj{t) = [Odi{t) ^d^]^ in the joint-coordinate systems are also given by e j j ( i ) = atsai2{y,x)

~ g,ttin2{k,x'^

+ y^ + /? -

l\)

0d2{t) = atan2(«;,a;^ + y"^ - l\ ~ ll)

K = yj{x^ + y^ + q + qy - 2[{x^ + y^y + it + it]

(20) (21) (0-7)

T h e reference angular rates are assumed to be ed{t) = [^di(t) 6d2]'^ = 0. In the following simulations, note t h a t the nominal p a r a m e t e r s of Table 1 arc assumed t o be used for a controller design, b u t the actual values are used in the numerical model.

503

7.1 Fuzzy Compensation In this simulation, the fuzzy compensator was apphed to the computed torque control system. Five membership functions and the control rule table t h a t were applied to the simulation are shown in Fig.7 and Table 2, respectively. Here, each label means t h a t N B : negative big, NM: negative medium, ZO: zero, P M ; positive m e d i u m , and P B : positive big. T h e membership functions were also assumed to consist of discrete sets with the support set [—12,12]; they took the discrete grades [0,1.6,3.3,5.0,6.6,8.3,10]; and their values were stored as a look-up table in the computer. We first considered a case when we does not tune the feedback gains Kp = d\^g{kj,i,kj,2),Kv = diag(/;„i, ^£,2) to obtain the desired tracking error responses, in which the feedback gains t a b u l a t e d in Table 3 were determined by k\,i = 2\Jkj,t. T h e i n p u t / o u t p u t scalers for fuzzy controllers used here are also t a b u l a t e d in Tabic 4. T h e control results are shown in Figs.8 ~ 11, in which Fig.8 represents the results due to only a C T (computed torque) m e t h o d with u n t u n e d gains; Fig.9 stands for the results due to adding the F C (fuzzy compensator) to the C T method; Fig. 10 represents the results due to only a C T m e t h o d with tuned gains; and Fig. 11 is the work coordinate trajectory corresponded to Fig.9. It is seen from these figiu-es t h a t the performance of the C T controller with u n t u n e d gains is improved by adding the fuzzy compensator having adaptive input scalers. Table 2 Control rule table for 5 labels Ae,

NB NM ZO PM PB

NB

NM

ZO

PM

PB

-12 -6 -12 -6 6

-12 -6 -6 6 6

-12 -6 0 6 12

-6 -6 6 6 12

-6 6 12 12 12

7.2 Multiple Fuzzy Control In this simulation, we simulated a multiple fuzzy controller having two elemental fuzzy controllers F C l a n d F C 2 , in which the F C l has the control rule given by Table 2 and the FC2 has the control rule given by Table 5.

504

Table 3 Feedback gains of computed torque control

k„ [1/s^] fcm [1/s] k. . [1/s'] Link 1

10

3

*

Link 2 Fig.ll Link 1

40 30

6 8

* 150

Link 2

100

9

300

Fig.9

Table 4 I n p u t / o u t p u t scalers for fuzzy compensation

Link 1 Link 2

GIN(f)

GIN(2 + i)

GOU(0

1000 1000

10000 10000

0.03 0.00007

In the following, we assumed t h a t the weighting factors Wi are u p d a t e d every sampling, so t h a t the s u m m a t i o n in (11) or (14) is excluded, and t h a t the trial of one learning is m a d e within the control interval 10s. Note t h a t if the input torque is constant t h e n Auj{kT) becomes zero, so t h a t wc can not c o m p u t e (17). To overcome such a singularity, we modified (16) and (17) such as Wi(l -f 1) = I'^iJ) if Auj(fcT) is approximately zero. T h e results of a case when each elemental fuzzy controller is singly applied to the plant are shown in Fig.12, where i n p u t / o u t p u t scalers for F C l and F C 2 are t a b u l a t e d in Tables 6 a n d 7. It is seen from this figure t h a t the control results are not necessary t o be satisfactory. It is observed, however, t h a t the result due t o the multiple fuzzy controller gives a good control response after one learning trial as shown in Fig.13, where T] = 0.0001, tz;i(0) = W2(0) = 0.5. T h e associated work coordinate trajectory is also depicted in Fig.14.

505 T

1

1

I

1

1

1

55 0.8

1

5

0 ^1.2

1

r

Set point Response

1

1

1

1

1

1

T3 (0

10 1

1

1

1

1

,f^

^0.8 CD 3 3-04 " \ O 1 1

^ V

// // ^ //// // // // //

• \

\

\ -

I

l

1 ""

l

10

5 Time (s)

Fig.8 Control results due to the computed torque method with untuned feedback gains

~1

1

1

1

r

"1

I

"i 0.8

J

1?

1

I

]

5

0 T3

V

Set point Response

1

1

1

1

1

1

1

1

km . ^^-^ (M OR

1

+-• 3

o

1

\

O)

Q. 3

L

10

\ -

0.4 " 1

1

1 1

5 Time (s)

1

1"

10

Fig.9 Control results due to the computed torque inethod with fuzzy compensation

306 1—I—I—I—r

— Set point — Response

roO.8

f 0.4 5 Time (s) Fig.10 Control results due to the computed torque method with tuned feedback gains

1

1

1

400

1

1

1

- Desired trajectory - Actual trajectory

I 380

// // //

\

i \\

I ' 360 >-

1

ll

// // // // >>•

3401

260

1

1

280

1

1

1

1

300 320 X-Axis (mm)

1

1

340

Fig.11 Control result in work-coordinate systems due to the computed torque method with fuzzy compensation

507

Table 5 Control rule table for 3 labels

Ae,-

ei

NB NB ZO PB

ZO

PB

-12 -12 -12 12 -12 0 12 12 12

Table 6 Input/output scalers for fuzzy controller FCl GIN(i) GIN(2 + i) GOU(0 Link 1 Link 2

100 100

5000 5000

0.1 0.01

Table 7 Input/output scalers for fuzzy controller FC2

GIN(i) GIN(2 + 0 Link 1 Link 2

500 500

10000 10000

GOl]{i) 0.05 0.005

508 1

l i 0.005

1

1

1

r

CD

o I™

CD

g-O.OOSh

W

0.02

CD

m "3 -0.02h

f Time (s) Fig. 12 Control results due to the elemental fuzzy controllers FCl and FC2

1

2

I

I—I

I—I—I—I—I—1—r

0.004

f: -0,004 zi a. 3 -O.OOBi

0 th

-J

t

J

I

I

0 g

L

I

.J

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8. C O N C L U S I O N S We have applied the fuzzy logic controller for controlling a irmlti-link robot manipulator with rubber muscles. We usuallj can not obtain the perfect physical parameters to mathematically model the manipulator and we also require some trials and errors to tune the feedback gains of the computed torque controller, even though we have a considerably accurate model. Using the independent fuzzy reasoning, we have first shown that the simple fuzzy controller is useful for designing a robust computed torque control system. Kext, we have also shown that an iterative learning fuzzy controller is eiiective for controlling the same manipulator without any modehbased controller. 9. R E F E R E N C E S C h e n , Y.H., R o b u s t Control System Design: K o n - A d a p t i v e Versus A d a p t i v e , Int. J. Contro!, 51-6, pp.1457^1477 (1990) Corless, M . J . , Tracking Coiitrolleis for Uncertain Systems: Application to a Mantitec R3 R o b o t , ASME J. of Dt/namic Syaiems, Meas. and ConiToI, 1 1 1 , pp.609^618 (1,989) Corless, M.,1. and L e i t m a n n , G., Coiitiiiuoiis S t i t e Feedbaclt G u a r a n t e e i n g Uniform U l t i m a t e Boundedness for Uncertain Dynamical Systems, IEEE Trans. Aut. Control, A C - 2 0 , p p . H S 9 - H 4 4 (1981) Daley, S. and Gill, K . F . , A design s t u d y of a self-organising fuzjy logic controller, Proc. htsi. Meek. Eng., 2 0 0 - C l , pp.59^69 (1986)

510 DeCarlo, R.A., Zak, S.H., and M a t t h e w s , G.P., Variable S t r u c t u r e Control of Nonlinear Multivariable Systems: A Tutorial, Proc. IEEE, 7 6 , pp.212-232 (1989) Linkens, D.A. and Hasnain, S.B., Self-organising fuzzy logic control and application to muscle relaxant anaesthesia, lEE Proceedtngs-D, 1 3 8 - 3 , pp.274-284 (1991) Hayashi, I., Nomura, H., and Wakami, N., Acquisition of Inference Rules by Neural Network Driven Fuzzy Reasoning, J. of Japan Society for Fuzzy Theory and Systems, 2-4, pp.585-597 (1990) (in Japanese) Horikawa, S., Furuhashi, T., O k u m a , S., and Uchikawa, Y., A Learning Fuzzy Controller Using a Neural Network, Trans, of the Society of Instrument and Control Engine€rs,27-2, pp.208-215 (1991) (in J a p a n e s e ) Hui, S. and Zak, S.H., R o b u s t Synthesis for U n c e r t a i n / N o n l i n e a r Dynamical Systems, Automatica, 28-2, pp.289-298 (1992) Imura, J., Sugie, T., and Yoshikawa, T., A d a p t i v e R o b u s t Control for R o b o t M a n i p u l a t o r s , Trans, of the Society of Instrument and Control Engineers, 2 7 - 3 , pp.314-319 (1991) (in J a p a n e s e ) Maeda, M. and M u r a k a m i , S., Self-Tuning Fuzzy Controller, Trans, of the Society of Instrument and Control Engineers, 24-2, pp.191-197 (1988) (in J a p a n e s e ) M a m d a n i , E.H., Application of fuzzy algorithms for control of simple d y n a m i c plant, Proc. lEE, 121-12, pp.1585-1588 (1974) M a m d a n i , E.H., Advances in t h e Linguistic Synthesis of Fuzzy Controller, Int. J. Man-Machine Studies, 8-6, pp.669-679 (1976) M i z u m o t o , M., Fuzzy Reasoning M e t h o d s for Fuzzy Control, J. of the Society of Instrument and Control Engineers, 2 8 - 1 1 , pp.959-963 (1989) (in J a p a n e s e ) Procyk, T . J . and M a m d a n i , E.H., A Linguistic Self-Organizing Controller, Automatica, 1 5 , pp.15-30 (1979) Psaltis, D., Sideris, A. and Y a m a m u r a , A.A., A Multilayercd Neural Network Controller, IEEE Control System Mag.,, 8, pp.17-21 (1988) R u m e l h a r t , D.E., McClelland, J.L., and the P D F Research G r o u p , Parallel Distributed Processing, 1, M I T Press (1986) Tanji, J. and Kinoshita, M., A Fuzzy Controller with a R o b u s t Learning Function, T r a n s , of ikt Society of Instrument and Control Engineers, 23-12, pp.1296-1303 (1987) (in J a p a n e s e ) Utkin, V.L, Variable S t r u c t u r e Systems with Sliding Modes, IEEE Trans. Aut. Control. AC-22, pp.212-222 (1977) W a t a n a b e , K., Fukuda, T. and Tzafestas, S.G., Learning algorithms of layered neural networks via extended K a l m a n filters, Int. J. Systems Set., 22-4, pp.753-768 (1991) W a t a n a b e , K. and Tang, J., Learning Controller based on Fuzzy Gaussian Neural Network, Proc. of 2nd Symposium, on Intelligent Systems, Nagoya, 920-87, pp.255-260 (1992) (in J a p a n e s e ) W a t a n a b e , K., Shiramizu, K., and Fukuda, T., Multiple Fuzzy Controls, Trans. JSME, Series G, Vol.58, No.554, pp.2970-2976 (1992) (in J a p a n e s e ) W a t a n a b e , T. and Ichihashi, H., Fuzzy Control of a Robotic M a n i p u l a t o r by the Feedback Error Learning, Trans, of the Institute of Systems, Control and Information Engineers, 3-7, pp.212-217 (1990) (in J a p a n e s e ) Yamazaki, T. and Sugeno, M., Self-Organizing Fuzzy Controller, Trans, of the Society of Instrument and Control Engineers, 20-8, pp.720-726 (1984) (in J a p a n e s e )

CHAPTER 19

FUZZY PETRI NETS AND APPLICATIONS Carl G. Looney Computer Science Department/168 University of Nevada Reno, NV 89557, U.S.A.

I. INTRODUCTION Networks for information flow were invented in the dissertation of C. A. Petri (1961) [21] at the University of Bonn. A. W. Holt and F. Commoner (1970, [9]) extended them and called them Petri nets. Peterson (1981, [20]) and Reisig (1985, [23]) provide treatises on the theory of Petri nets, that are used nowadays to model flows such as parts and materials in manufacturing processes (Zhou and Leu, 1991 [32]), and logical truths in machine reasoning systems (Looney, 1988, [15], and Murata et al, 1991, [19]). For our purposes, a logic Petri net P consists of i) a net architecture; and ii) an operational procedure. Architecturally, it is a directed graph that consists of two kinds of nodes: a) conditions, designated by circles; and b) events, denoted by bars. The conditions and events are connected by arrows according to two rules: 1) an arrow may connect from a condition to an event, or from an event to a condition; and 2) an arrow may never connect two nodes of the same kind. Figure 1 is an example of such a scheme. Procedurally, the net uses tokens, represented graphically by dots inside condition nodes, to activate the conditions where they appear. A token denotes a Boolean truth value of 1 for the condition, while the absence of a token indicates a value of 0. An event is enabled when every arrow entering it comes from a condition that contains a token. An enabled event fires to activate (i.e., make true) all conditions to which its departing arrows directly connect, by sending them tokens. In Figure 1, Condition C2 is activated, i.e., made true, with a token. A clock is implicit in the net in that it connects to each event so that upon being enabled, an event fires on the next clock pulse. A clock is necessary to maintain sequential order. A logic Petri net that interacts with the external world has boundary nodes that connect to the external environment. In Figure 1, Condition Cj, and Events Ej, and E^, are boundary nodes. Arrows enter boundary conditions from the environment, and arrows depart boundary events to the environment. External events that activate boundary conditions are called sources, while external conditions that are activated by boundary events are called sinks. The operation of a logic Petri net is initiated when certain source events fire to activate boundary conditions. These enable events that fire to activate other conditions, which may in turn enable other events to fire to activate other conditions, etc. 511 S.G. Tzafestas and A.N. Venetsanopoulos (eds.) Fuzzy Reasoning in Information, Decision and Control Systems, 511-527. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.

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Conditions are propositions, i.e., statements that must be either true or false. In Figure 1, conditions Cj, Cj and C^Q are true initially, as designated by the tokens. This causes Event Ej to be enabled by Cj and Cj, so that Ej fires on the next clock pulse to activate

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conditions C3 and C4, as shown in Figure 2. The truth of C4 enables Event E3, which fires to activate Condition C^, and C3 enables Ej, which fires to activate both conditions C5 and Cg. Further enabling, firing, and activation cause Event E5 to fire to propagate a truth token to a sink. Note that there is no conservation of tokens, which may split or combine. The firing of events corresponds to the modus ponens syllogism. Figures 1 and 2 show that the net architecture contains the built-in rule [(Cj AND Cj) - • (C3 AND C4)], which is equivalent to the two rules [(C^ AND Cj) -» C3] and [(C^ AND Cj) "* C4]. When the rule antecedent (C^ AND C2) is activated (made true), then the rule events (implications) are enabled to fire and activate the rule consequents C3 and C4. Thus AND logic can be modeled in both antecedents and consequents of rules. Condition Cg can be activated by the firing of either Event E3 or Ey, so OR logic is implemented. We incorporate NOT into the net by putting a small circle on the end of an arrow to denote negation, as shown at Event E7 in Figures 1 and 2, to signify the rule [NOT CJQ - • Cg]. Nonspecific rules of the form [C^ - • (C^ OR C„)] are not modeled because they do not specify which of the consequent conditions is activated. Rules of the form (C OR C^) -* Cn, may be considered to be two separate rules: [Cj -» C^], and [C^ -» C^]. Now let C = {Cj,...,C„} be the conditions of a logic Petri net P. A marldng of P is a vector M = (mi,...,mj that distributes truth values over the n conditions, where raj = 1 denotes a token (truth value of 1) for Condition Cj, else m; = 0 indicates that C; is false. We also call M the truth state of P. In Figure 2 the truth state is M = (1,1,1,1,0,0,0,0,0,1). Because a Petri net is dynamic, there is particular interest in any sequence {Mo,Mj,...,Mf} of markings that ensues from firings of events due to an initial marking MQ. We say that a marking Mj is reachable from MQ if and only if there is a sequence of such markings MQ,...,Mf. When an initial marking M^ starts a sequence of markings (truth states), the net may eventually reach a marking Mj that does not change further. In such a case, Mj = (mj^^,...,m„^^ is the final truth state and indicates all of the true conditions that can be deduced from the initial truths in MQ. Deadlock is a situation where the logic Petri net has no enabled events, in which case no rules can fire to propagate truths. A deadlocked Petri net remains in a fixed state (marking). n . FROM FUZZY SETS TO FUZZY VARIABLES Fuzzy control arose in response to difficulties in the control of complex systems. According to D'Ambrosio (1989) [1]: i) there may be no detailed model of the process to be controlled, ii) known models may be too complex, nonlinear, or time-varying to be handled by tractable mathematical methods, iii) necessary assumptions may be unacceptable, iv) the derived models may not respond adequately to noise in certain of the variable measurements, and v) the performance and stability may be unsatisfactory. Mamdani (1974) [16] applied Zadeh's ideas (1965 [30], and 1973 [31]) to the problems of control, and thereby initiated fuzzy control with qualitative fuzzy rules that emulate an expert human operator using heuristic rules (see Mamdani, 1975, [17], and 1976, [18]). Tong (1977) [28] described the logic for fuzzy rules in fuzzy control systems, and Sugeno (1985) [26] provided an overview of fuzzy control. Expert systems are rule-based systems restricted to a particular domain of expertise (see Hayes-Roth (1985) [7]). Efistathiou (1987) [3] applied rule-based systems to industrial control. Looney (1988) [14]

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used fuzzy truth state operated on by fuzzy rule matrices with MIN/MAX logic to reason forward, and also used a fuzzy Petri net (1988) [15] to model fuzzy reasoning for control. Lipp (1983) [13] also used a Petri net for modeling a complex controller, but it did not implement logic flow. Postlethwaite (1990) [22] discusses fuzzy expert systems. A fiazy set F (with respect to a universal domain set X) is the set of all elements x e X, along with a membership Junction fif(x), where 0 « ^p(x) s 1, that gives the extent to which each x in X belongs to F. For fuzzy sets F and G on the domain X, the fiazy intersection is the set FDG of all elements in X with the fuzzy membership function defined via: fifpfjOi) = min{Mp(x),/Zo(x)}. Thus x belongs to FflG only to the (least) extent that it is a member of both F and G. The fiizzy union is the set FUG of all elements x e X with fuzzy membership % U G W ~ ^^^{f^^)^(^)}Then x belongs to FUG to the (greatest) extent that it belongs either to F or G. The fuzzy complement of F is ~F with fuzzy membership function yM_p(x) = 1 - fif(x), so that x belongs to ~F to the extent that it does not belong to F. When let X and Y be two different universes of discourse, and F is a fuzzy set on X and G is a fuzzy set on Y, then what is the meaning of FflG and FUG? Because / ^ x ) and ^ ( y ) each take on values between 0 and 1, they are comparable, so we define E = FOG to be the fuzzy set on XxY that has set membership function fij:fya{^,y) = m[n{/if(x),fiQ(y)} for any (x,y) E XXY. This is the extent to which it is true that x belongs to F and y belongs to Y. Similarly, H = FUG is a fuzzy set with /HpjQix,y) = We adopt a nonstandard perspective that a fuzzy set F is a fiazy variable, analogous to a Boolean variable. While a Boolean switch is either OFF or ON (0 or 1), a Zadeh (fuzzy) switch can be any proportion of the way between OFF or ON. A fuzzy variable F that represents a condition has an associated fuzzy truth value |F|. If F is the condition, "Temperature is MEDIUM," then |F(x)| = |Up(x) provides the extent to which the temperature belongs to the fuzzy set MEDIUM, defmed on the range of temperature values of X (this fiazifies the temperature readings x). Now let G be the fuzzy variable for the proposition, "The valve is OPEN." Then ~G is the proposition for, "NOT the valve is OPEN", or, "The valve is CLOSED." If |G(x)| = 1/4 (the valve is 1/4 of the way open for x), then |~G(x)| = 1 - |G(x)| = 3/4 (the valve is 3/4 of the way closed). The AND and OR logical operations of Boolean logic can be defmed, respectively, as MIN and MAX (for minimum and maximum), which is included in the fuzzy case. The NOT Boolean operation is also included in the fuzzy complement ~F = 1 - F. n i . FUZZY LOGIC SYSTEMS We can now define a fiazy logic as a triple J ^ = (V,T,IH), where V = {A,B,C,...} is a set of fuzzy variables, T = [0,1] is an interval of truth values, and "H - {MAX, MIN, NOT, -», • } is a set of logic operations on VxV into V (or V into V for the unary NOT). Table 1 displays the truths of the mapped variables. The first row is MIN (AND), the second is MAX (OR), the third and fourth apply to implication (-^), the fifth is equivalence (•), and the sixth is NOT. If A and B are Boolean variables, then they are restricted to the values 0 and 1, in which case MIN (AND), MAX (OR) and NOT hold. Fuzzy equivalence includes the Boolean case, but the fuzzy case allows a fuzzy degree of equivalence.

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INPUT LOGIC VARIABLES 1. 2. 3. 4. 5. 6.

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Implication is a combinational operation that assigns a fuzzy truth to the result, and is not the same as the modus ponens law of logic [A AND [A -»(f) B] that means: the implication fuzzy truth f of [A -* B] and the fuzzy truth |A| of A imply the fuzzy truth |B| of B. The fuzzy truth f is a proportion of openness of a fuzzy switch, so an implication [A -»(f) B] with truth |A -»(f) B| = f > 0, and the fuzzy truth |A| of A, imbue B with the truth of A up to the truth (openness) f of the implication switch. On the other hand, the implication truth f can not pass a truth to B that is greater than the source truth of A. Therefore, the model for the truth of B is |B| = min{|A|,f} = |A| MIN (f). This model also holds for implication in Boolean logic. Of course, B may have a truth value due to some other implication or due to fuzzification. We may restrict the implied truth of B to the A-context, i.e., the truth contribution to B only from A, designated by jB^^^J. Contextual fuzzy implication holds for Boolean logic also. Fuzzy modus ponens: This is the basic implication law: (A AND [A -»(f) B]) - • B. The transferred fuzzy truth due to A and [A - • B] is defined to be |B/^J = |A| MIN f, according to the above discussion. A must be initialized with a positive truth value, i.e., become a fUzzy fact, to fire and activate B with a fuzzy truth value. Fuzzy modus ponens and contextual implication allow the building up of evidential truth for B from the various contextual implications, and then combining all truths logically for a final truth value, say |B| = max{|B,y^-,|,|B,D-,|}, where A and D each implies B. For example, let (A ^•(0.3) B) and (D -»(0.8) B) be propositions. For |A| = 0.6 and |D| = 0.7, B^^) is activated (only from A) with the truth min{0.6,0.3} = 0.3, but B-^s is activated (only from D) with the truth min{0.7,0.8} = 0.7. Thus the truth of B fi-om A OR D is |B| = max{|B, J.lD,^)!} = max{0.3,0.7} = 0.7. Not all laws of Boolean logic extend to fuzzy logic (see Sugeno and Takagi, 1983 [27], Dubois and Prade, 1987 [2], and S. Weber, 1985 [29]). Kosko (1986) [10]) questions the truth of the law of contraposition (modus tollens) in the world of partial (fuzzy) implication (i.e., (A - • B) is true if and only if (~B - • ~A) is true). His example is that (smoking -» lungcancer) is true to some extent, but that does not imply the extent to which (NOT lung_cancer -* NOT smoking) is true. We demonstrate that fuzzy contraposition holds under constraints. Fuzzy Contraposition: Let |A| s f and 1-|A| s f. Then X = (A -»(f) B) has fuzzy truth f if and only Y = (~B -•(f) ~A) has fuzzy truth f. For the Boolean case where f = 1, this always holds. We note that this is not the fuzzy modus tollens logic law: (NOT B AND [A - • B]) -^ NOT A).

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Proof: Let |A| s f and |~A| = 1-|A| s f. Then X = (A -»(f) B) means that |B| = min{|A|,f} = |A|. Thus 1-|B| = 1-|A| so that |~B| = |~A|. Then we can write (~B -•(f) ~A), because |~A| = |~B| = min{|~B|,f}. By symmetry due to ~(-F) = F, the argument in the reverse direction holds.' For a counterexample, let |A| = 0.2 and f = 0.6, so (A -*[0.6] B) means that |B| = min{0.2,0.6} = 0.2. But |~B| = 0.8, so that (~B -*[0.6] ~A) means that |~A| = min{0.8,0.6} = 0.6 ^ 0.8 = 1 - 0.2 = 1 - |A|. Thus contraposition fails here, and f < 1 |A|. For another counterexample, let |A| = 0.75 and f = 0.5, so that |B| = min{0.75,0.5} = 0.5. Then contraposition (modus tollens) means that |~A| = min{|~B|,f} = min{0.5,0.5} = 0.5 K = 0.25 = 1 - 0.75 = 1 - |A|. For an example that satisfies the hypothesis of fuzzy contraposition, let |A| = 0.4 and f = 0.8. Then |B| = min{0.4,0.8} = 0.4. Contraposition then implies that |~A| = min{|~B|,f} = min{ 1-0.4,0.8} = 0.6, so that |A| = 1 - |-A| = 0.4, which is consistent. The fuzzy operations NOT, MIN, MAX, contextual implication, and equivalence are sufficient for our needs, and they imply the truths of certain laws such as the De Morgan laws, which are needed to convert to conjunctive normal form (CNF) or disjunctive normal form (DNF) (see Schalkoff, 1990 [25], or Rich and Knight, 1991 [24]). The former form is the conjunction (ANDing, or MINing) of disjunctions (ORing, or MAXing, of variables), e.g., (A MAX B) MIN (C MAX D MAX E) MIN (F). The latter form is the disjunction (ORing, or MAXing) of conjunctions (ANDed, or MINing, of variables), e.g., (A MIN B MIN C) MAX (D) MAX (E MIN F). These are also known as productof-sums and sums-of-products, respectively. Fuzzy De Morgan Laws: i) ~(X MIN Y) = (~X MAX -Y); ii) -(X MAX Y) = (~X MIN ~Y). Proof: The first one follows from the fact that ~(X MIN Y) has the truth value 1 min{X,Y} = 1 - (1 - max{l-X,l-Y}) = max{l-X,l-Y} = (~X MAX ~Y). The second argument is similar.' IV. FUZZY RULE-BASED SYSTEMS R. C. T. Lee (1972, [11]) investigated the principle of resolution in fuzzy logic. Forsyth (1984) [4] discusses expert systems with fuzzy values providing certainty factors. Recently, Garg et al (1991, [5]) addressed the question of reducing fuzzy clauses to check consistency in a fuzzy knowledge base. Informally, a fuzzy proposition is any statement that can be assigned a fiizzy truth value. Formally, a Jiizzy proposition is: i) any fuzzy atom (an indivisible fuzzy proposition); ii) any result of MINing or MAXing two fuzzy propositions; iii) the complement of any fuzzy proposition; iv) any contextual implication (A -*(T) B) for any fuzzy propositions A and B; and v) any finite combination of the above. We will use fuzzy variables to represent propositions. Afiizzy rule is a fuzzy proposition of the form X = (A -»(f) B), where A and B are fuzzy variables that represent fuzzy propositions (conditions) that assume fuzzy truth values |A| and |B|. A and B are, respectively, the antecedent and consequent propositions, or conditions, of the rule X. A fiazy knowledge base is a set of fuzzy propositions 9 that contains both a subset C of conditions and subset 91 of fuzzy rules connecting the conditions to model the behavior of a system, or environment. The rules have nonzero

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fuzzy implication truths. Conditions may have zero truth values until initialized with fuzzy truths for a process of fuzzy deduction, at which time they become fuzzy facts. Others may obtain nonzero fuzzy truths upon being activated by the firing of one or more rules. Aft4zzyrule-based system contains: i) a fuzzy knowledge base; ii) a condition initializing function to post fuzzy facts; iii) an inference function for scheduling and interpreting rules (to fire each rule or not); iv) a user interactive procedure; and v) an editor. V, FUZZY PETRI NETS A fuzzy Petri net is a logic Petri net that uses fuzzy logic rather than Boolean logic. The architecture softwires a specific set of rules of the forms: i) (Aj AND...AND A^) - • (Bj AND B2 AND...AND B„); ii) (Ai OR Aj OR...OR A„) - • (Bj AND...AND B„); iii) [(All AND...AND Ai J OR...OR (A^i AND...AND A^.^] — (Bj AND...AND B„); or iv) [(All OR...OR Aj J AND...AND (\i OR...OR A^p)] -* (Bi AND Bj AND...AND B„); for k, m, n * 1, where the Aj and Bj are fuzzy variables that represent conditions. Of course, we use MIN in place of AND and MAX in place of OR. As in the case of the logic Petri net, we do not use nonspecific rules that have ORed consequents. We also break up ORed antecedents to form separate rules, similar to Prolog. For example, ((A OR C) -» B) becomes (A - • B), (C -»• B). Figure 3 shows a fuzzy Petri net. FIGURE 3 - A SIMPLE FUZZY PETRI NET

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518

the postset N; of Nj. For example, in Figure 3, Ej = {€^,€5} is the preset of E^, Ej {Cg} is the postset of Ej, and Cj^* = {Ej.Es} is the postset of CJQ. FIGURE 4 - EXAMPLES OF ACTIVATING AND MARKING UPDATES (a) Example o f nj

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The Event Ej of Figure 4(a) is enabled only if each condition in its preset Ej contains a token that represents a nonzero truth value (a nonzero fuzzy marking). These fiizzy values from the preset conditions are MINed to obtain a fuzzy enabling value Cj = min{mi:Cj£*Ei} at Ej. Because an Event Ej represents an implication, it must have a fiizzy implication value fj. The activating value propagated across the implication event is then a^ = min{fj,ej} = min{0.65,0.6}, and this activates each of the conditions in E^' with the fuzzy truth aj = 0.6. In Figure 4(a), the Event E^ enabling value was obtained via Bj = min{mi,m2,m3} = min{0.6,0.7,0.8} = 0.6. The Condition C^ of Figure 4(b) is activated by more than one event in its preset C^. Each Event E- e C4 activates C4 with an activating value a: that provides evidence for the truth of C4 only from Ej (contextual implication). Another Event E^ e *C4 also activates C4 and provides its contextual implication (evidence) a,^ only from E^^. But the existing truth (fuzzy marking) of C4 also represents evidence obtained previously from some other origin. Then the updated activating truth, or evidence, for C4 is 34 = max{|C4|,a^aj} = max{m4,a,j,aj}. On the clock pulse where a4 becomes the fuzzy truth of Cj, then a4 becomes the new marking (state) m4 for C4, where j = 1 and k = 2 in Figure 4(b) that shows the computation of a new marking m4 for Condition C4, where the present fuzzy truth of C4 is m4 = 0.7. The new marking is assigned a new fuzzy value via m4 «max{aj,a2,m4} = max{0.6,0.76,0.7} = 0.76. A thresholded fuzzy Petri net uses a fuzzy activation threshold value X- at each jth Event Ej. When Event Ej with threshold t has a fuzzy implication value f, and an

519

enabling value e,, then the enabled Event E: fires only if the activating value aj = min{£,ej} satisfies aj s tj. Thresholds guarantee that decisions are not made based on weak evidence. VI. A FUZZY PETRI NET ALGORITHM The fuzzy truth state of a fuzzy Petri net with n conditions Cj .Cj,—, C„ is the current vector of fuzzy markings M = (mi,m2,—,m„), where m-^ is the current fuzzy truth of Cj. Let E = (ej,e2,.".e„) be the vector of event enablings, where the Event Ej has the enabling value Cj given by the minimum of all marking values (activations) of its preset conditions in Ej. An algorithm to propagate the initial fuzzy truths of the conditions through the system needs two stages: i) for each Event Ej, j = 1 m, get the markings of the conditions in the preset Ej, compute the event enabling value Cj by minimizing these markings, and then compute the activating value aj = min{ej,fj}; and ii) for each Condition Cj, get the activating values of its preset events in 'Cj and compute the new marking m as the maximum of the current marking mj of Cj and all activations from preset events. An algorithm requires data structures. For n conditions and m events, EP[j,i] is the event preset matrix, where the jth row represents the jth Event Ej, 1 s j s m, and the ith column, 1 s i s n, of the jth row holds the marking from the preset Condition Cj E *EJ. For example, EP[2,3] = g denotes that Ej has preset Condition C3 with current marking mj = g. Then the enabling value Cj for Ej can be computed via Cj = min{EP[2,i]: 1 s i s n}. The activation aj from E2 is then determined to be aj = min{e2,f2}Similarly, we define CP[i,j] to be the condition preset matrix, I s i s n , I s j s m . Then CP[ij] = g means that Condition C^ has activation value aj = g = CP[i,j] from Event Ej E Cj. Then the new marking for Cj is computed via mj = max{mi,aj: 1 s j s m}, where aj = CP[i,j] for each j . Note that any external conditions that can be activated by boundary events must be listed in the data structure so they can be updated, and any external events that feed into boundary conditions must be listed, and updated by an external function. We present here the high level description of the algorithm. A Fuzzy Petri Net Algorithm: Inputs: {data structures EP[j,i], 1 s j s m, 1 s i s n; and CP[i,j], 1 s i s n, 1 s j s m; initial marking M^ = (mj m^); fuzzy implication vector F = (fj,...,f^)} Outputs: {current marking M = (mj,...,mn), current enabling vector E = (ei,...,e„), current activation vector A = (ai,...,a^)} Step Step Step Step Step Step Step

0: 1: 2: 3: 4: 5: 6:

Initialize (EPO.i], CP[i,j], F, M„) Update_Extemal_In /* update Update_Enablings /* update UpdateActivations /* update Update_Markings /* update UpdateExtemalOut /* update IF (Stopping_Criterion) THEN Exit ELSE Goto_Step_l

inputs to boundary conditions */ enabling values C: */ activating values aj */ markings mj */ outputs from boundary events */

520

Any external fuzzification and defuzzification may be done by the Update_Exteraal_In and UpdateExternalOut functions. The functions UpdateEnablings, UpdateActivations, and UpdateMarkings update the enabling, activating, and marking vectors, respectively. We present lower level procedures for UpdateEnablings, Update_Activations, and Update_Markings. Procedure Update_Enabiings: FOR j = 1 to m DO /* m events to be enabled */ emin = 1.0; FOR i = 1 to n DO /* minimize up to q preset conditions */ IF (EP[j,i] s emin) THEN emin = EP[j,i]; e[j] = emin; /* enabling value is minimum value */ Procedure Update_Activations: FOR j = 1 To m DO IF (fjj] < elj] THEN aU] = f|j] /* min. result with fuzzy implication */ ELSE a[j] = e[j] /* to obtain fuzzy activation */ Procedure Update_Markings: FOR i = 1 to n DO /* n conditions to be updated */ amax = 0.0; FOR j = 1 to m DO IF (CP[i,j] a amax) THEN amax = CP[i,j]; IF (amax i m[i]) THEN m[i] = amax; /* maximum of old marking*/ /* and amax */ Vn. TRAINING FUZZY PETRI NETS AS DECISIONMAKERS A fuzzy Petri net is a model for decision making via a fuzzy rule-based system. It is not a traditional fuzzy controller. We simplify the fuzzification and defuzzification processes by using proportional mappings. For example, the four inputs of Figure 3 could be temperature, pressure, volume, and velocity, each of which has a range of the form [a,b], where a is the lower bound and b is the upper bound. The linear map x -» y = (x-a)/(b-a) takes a into 0, b into 1, and x that is p of the way between a and b into p. The fuzzy outputs also represent proportions that can be mapped back into a range [a,b] via: y -*• y(b - a) + a. A nonlinear squashing function could also be used, such as the sigmoid x - • 1/[1 + exp(-ax+6)], or x - • V^[l + tanh(ax)]. Instead of fuzzy sets such as LOW, MEDIUM, and HIGH, this implements a continuum. The fuzzy Petri net designer must have available certain rules that model the qualitative behavior of a process to be decided, e.g., a plant to be controlled. The net schematic is drawn to represent these rules, and then softwired into data structures by inputting the matrices EP[j,i] and CP[i,j] that represent the architecture of the fuzzy Petri net. The fiizzy implication values F = (fj,...,f^) for the events need not be known: they may be initially set to 0.5 or to subjective coarse values. The fuzzy Petri net can be trained to adjust these for near optimal performance.

521

We consider here the case of supervised training of a fuzzy Petri net for control decisions. A time trajectory of desired plant outputs must be known. Figure 5 shows a general trainable fuzzy controller. For output variables, e.g., y and z, a given desired output {target) trajectory over time k is denoted by {(yd(k),Zj(k))}|j, while the actual plant outputs due to fuzzy Petri net decisions are designated by {(ya(k).ZaO'))}k- ^ ^ ^^"'^' squared error (SSE) is then computed over time k = 1,...,K as

E = 2(k=i.K) lya(k) - yd(k)P + k(k) - zM"FIGURE 5 - A GENERAL TRAINABLE FUZZY RULE-BASED CONTROLLER Plant State

Target MMSE I

TRAINER ^

1 • da

^

1

\

- ^onnanc ^ FUZZY " EXPERT CONTROLLER <

PLANT

•

Plan-t State

The process of training entails an initial computation of the SSE using the coarse fiizzy implication values. Then each single fuzzy implication value fj in turn is perturbed and the SSE is computed to determine whether or not the SSE decreases. If so, the new £ is kept, else it is discarded. This continues over all events Ej, 1 s j s m. When the SSE stops decreasing for a particular fj, then the process is repeated for the fuzzy implication fj+j, etc. When all fuzzy implications iy,...,ij^ have been adjusted to approximately minimize the SSE E, the training is done. The Fuzzy Controller Training Algorithm: Inputs: {Petri net architectural data structures EP[j,k], CP[i,r]; initial fuzzy implication vector FQ = (fi^,...,f^^; initial marking Mo = (m^^ m„J; sequence t(l) t(K) of target states, where t(k) = (tj(k),...,tj(k)); 6Q = fuzzy implication increment, stop criterion e, and Max_Loop_Count} Outputs: {actual plant states s(l),...,s(K); total sum square error E over target sequence; and final fuzzy implication vector F^}

522

Step 0: input initial fuzzy implication vector FQ; input target trajectory {t(k)}; input initial marking MQ; input EP(j',i]; input CP[i,j]; Step 1: input stop criteria s; input Max_Loop_Count; input fuzzy increment delta 6Q Step 2: compute the SSE E = S^^^i K) ^(i=\^) I tj(k) - s^Ot) I ^ Step 3: REPEAT /* repeat minimization until change is small */ LoopCount = 0 FOR j = 1 to m DO /* for each jth event, minimize f */ 6-5o count = 0 REPEAT fnewo) "^ fj + 6 such that 0 s f„^^j s 1; compute SSE E „ ^ with f^ IF (E„^ > E) THEN 6 = •-6*0.96; count = count + 1 ELSE E *- E „ ^ ; f. *- f^,,^,; 6 = 6*0.96; UNTIL count = 6 increment Loop_Count UNTIL (I E^oij - EI < £) OR (Loop_Count > Max_Loop_Count) If the rules are partitioned into a binary tree of disjoint cases with the rules that involve the single fuzzy implications f as leaf nodes, then we say that the rules are orthogonal. In this case each f need be minimized over a single turn (adjustments of other fuzzy implications f^ (k 4= j) do not affect it). Under feedback noise, the net deals with behavior in an average sense, and is equivalent to the minimum mean square error methodology, invented independently by Gauss (1809) [6] and Legendre (1810) [12], and implemented frequently nowadays (see any textbook on least mean-squares filtering, e.g., Haykin, 1991 [8]). The inner loop adjusts each fuzzy weight to lower the overall SSE. To obtain near-optimal fuzzy implications, the target trajectory must be sufficiently varied to invoke all rules so each can be adjusted. This process can be extended to also adjust the fuzzy values given to the conditions, based on plant feedback over k = 1,...K. Vni. A FUZZY LOGIC RULE-BASED CONTROL APPLICATION A fuzzy rule-based mapping is appropriate when some knowledge is known about the qualitative input/output behavior of a nonlinear plant. No mathematically tractable model need be known for the inner workings of the system. We present here a simple example where the control of a train is modeled by a fuzzy Petri net. In this example, Metrocity has two major downtown areas called North Center and South Center. Two passenger trains of a few cars each are to travel simultaneously in the same counterclockwise direction on an oval shaped track that connects the two centers. Train 1 will leave the South Station at the same time that Train 2 departs the North Station. Figure 6 shows the situation. The system is to satisfy the following specifications.

523

FIGURE 6 - AN EXAMPLE OF A TRAIN SYSTEM TO BE CONTROLLED NORTH CENTER Platform • Station N 1 \ - A p p r o a c h Sensor 7 \ - 7 / 8 - V a y Sensor —/

\= 3 / 4 - V a y

-

= l / 2 - V a y Sensor V Dne-Woy Rail Line

=\

/~ 1 / 4 - V a y

Sensor

Sensor

-\ J Direction o f PlQtforn Q Station S SOUTH CENTER

Trains

METRDCITY

1. Both trains are to start at the same time when the Doors_Secure, OtherTrainReady, Emergency, and CIock_Start signals are appropriate, which causes the Start signal to enable and activate the application of power increments until the train reaches the 1/4-way point, provided that the Emergency signal is 0.0. 2. At each of the half-way and 3/4-way points, there are to be two sensor strips a short distance apart for the purposes of sensing the positions and ascertaining accurate velocities. The speed of the trains will then be adjusted up or down toward the desired maximum speed. 3. The Emergency signal is to take two values: i) 0.0 means that there is no emergency; and ii) 1.0 means that an emergency exists. In the first case, all applications of power, and other functions, are enabled. An emergency automatically applies the brakes at the full nonlocking level. An emergency disables all applications of power. 4. At the 7/8-way points, power is reduced to approximately half-power. 5. At the Approach points, power is reduced to zero level (the trains coast). 6. At the Station-Entry points, brakes are applied at 1/2 level. 7. At the Platform-Edge points, brakes are applied at full nonlocking level. 8. When stopped, all fuzzy logic values are reset for the next trip, and the clock is started for a two-minute wait. Figure 7 shows a fuzzy Petri net that is a simplified high level model. When the Emergency signal at Node C is set to 0.0, it outputs a fuzzy value of 1.0 to enable power conditions. It and Nodes B, D, and E enable the event between B and I so that power incrementation is activated at Node I to enable a sink event. As long as the 1/4-way point has not been reached. Node F will be 0.0, and will output a 1.0 value to enable further increases in power. After the 1/4-way point is reached, the power increase by Node I is

524

FIGURE 7a - FUZZY PETRI NET TRAIN CONTROLLER

FIGURE 7b - FUZZY PETRI NET TRAIN CONTROLLER continued [Continued] Enei-g.

/AA

BOUNDARY

2)

CContinued in Next Frane]

disabled per the event between Nodes I and J. When the half-way point is reached, L is activated with fuzzy truth value 1.0, which makes it possible for the power to be increased at N or decreased at Q, depending upon what fuzzy values the Speed_Too_High or Speed_Too_Low signals take from the sensing of the speed at the 1/2-way sensing

525

Strips. The fuzzy values provided to one of these two signals determines the value to which the power will be changed, because each is MlNed with values of 1.0 of the 1/2way signal. FIGURE 7c • FUZZY PETRI NET TRAIN CONTROLLER continued [Continued]

Power

Dff

The fuzzy sensor values come in from the external world, while the fuzzy power level controls are sent out to the external world when a decision is made. At the 3/4-way point, the power level is adjusted again by a fuzzy value from Node CC, depending upon whether the speed is too high or too low, provided that there is no emergency. At the 7/8way point, the speed is adjusted to half speed in the absence of the Emergency signal. When the Approach signal is activated (and the Power signal is present from Node GG, then the PowerOff signal is activated at Node JJ. When the Station point is reached, the brakes are applied, at the fiizzy value 1/4, and then applied at fuzzy value 1/2 at the Platform signal. When stopped, the logic is cleared and the two-minute clock is started. When an enabled event fires, its enabling value is consumed, and that event must be enable again if it is to fire again. When an event is enabled by fuzzy tokens in its preset conditions, those tokens are consumed in the enabling. The control variables are usually fuzzy in order for proper adjustments to be made. It is obvious that a fuzzy Petri net model is not unique. For example, we have adjusted the power for speed change via two different structures: i) by Nodes N or M, one of which decreases and the other which increases. However, Node CC sets the power level through a new fuzzy value, determined either by DD or EE, that is either a decrease or increase from what it was. The designer may develop multiple versions before one is accepted as being advantageous in some manner, e.g., by being simpler.

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REFERENCES [1] B. D'Ambrosio, Qualitative Process Theory Using Linguistic Variables, SpringerVerlag, NY, 1990. [2] D. Dubois, and H. Prade, The management of uncertainty in fuzzy expert systems and some applications, in Analysis of Fuzzy Information, Vol. II, ed. by J. Bezdek, CRC Press, Boca Raton, 1987, 39-58. [3] J. Efstathiou, Knowledge-based systems for industrial control, Computer-Aided Engineering Journal, 4, 7-28, 1987. [4] R. Forsyth, Fuzzy reasoning systems, appeared in Expert Systems (Edited by Forsyth), Chapman and Hall, London, 1984. [5] M. L. Garg, S. I. Ahson, and P. V. Gupta, Fuzzy Petri net for knowledge representation and reasoning. Info. Proc. Ltrs., vol. 39, no. 3, 1991, 165-171. [6] K. F. Gauss, Theoria Motus Corporum Coelestium in Sectionibus Conicus Solem Ambientum, Hamburg (translation: Dover, NY, 1963), 1809. [7] F. Hayes-Roth, Rule based systems, ACM Comm. 28, 921-952, 1985. [8] S. Haykin, Adaptive Filter Theory, Prentice-Hall, Englewood Cliffs, 1991. [9] A. W. Holt, and F. Commoner, Events and Conditions, New York: Applied Data Research, also in Rec. Project MAC Conf. Concurrent Systems and Parallel Computation, New York: ACM, 1970, 1-72. [10] Bart Kosko, Fuzzy cognitive maps, Int. J. Man-Machine Studies 24, 1986, 65-75. [11] R. C. T. Lee, Fuzzy logic and resolution principle, J. ACM, vol. 19, no. 1, 1972, 109-119. [12] A. M. Legendre, Methodes des mondres quarres, pour trouver le milieu le plus probable entre les resultats de differences observations, Mem. Inst. France, 149-154, 1810. [13] H. P. Lipp, The application of a fuzzy Petri net for controlling complex industrial processes, IFAC Conf. Fuzzy Info. Contr., Marseille, Jul. 1983, 459-465. [14] C. G. Looney, Expert control design with fuzzy rule matrices, Int'l J. Expert Systems, vol. 1, No. 2, 159-168, 1988. [15] C. G. Looney, Fuzzy Petri nets for rule based decisionmaking, IEEE Trans. Systems, Man, and Cybernetics, Vol. 18, No. 1, 178-183, 1988. [16] E. H. Mamdani, Applications of fuzzy algorithms for control of simple dynamic plant, Proc. lEE, vol. 121, no. 12, 669-678, 1974. [17] E. Mamdani, and S. Assilian, An experiment in linguistic synthesis with a fuzzy logic controller, Int'l J. Man Machine Stud., 7, 1-13, 1975. [18] E. Mamdani, Advances in the linguistic synthesis of fuzzy controllers, Int'l J. Man Machine Stud., 8, 669-678, 1976. [19] T. Murata, V. S. Subrahmanian, and T. Wakayama, A Petri net model for reasoning in the presence of inconsistency, IEEE Trans. Knowledge and Data Engineering, vol. 3, no. 3, 1991, 281-292. [20] J. L. Peterson, Petri Net Theory and the Modeling of Systems, Prentice-Hall, Englewood Cliffs, 1981.

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[21] C. A. Petri, "Kommunikation mit Automaten," Schriften des Rheinish-Westfalischen institutes fiir Instrumentelle Mathematik an der Universtat Bonn, Heft 2, Bonn, W. Germany, 1962. [22] B. Postlethwaite, Basic theory and algorithms for fuzzy sets and logic, appeared in Knowledge-Based Systems for Industrial Control, Ed. by J. McGhee, J. Grimble, and P. Mowforth, Peter Peregrinus Ltd., London, 1990. [23] W. Reisig, Petri Nets, Springer-Veriag, Berlin, 1985. [24] Elaine Rich, and Kevin Knight, Artificial Intelligence, Second Edition, McGrawHill, NY, 1991. [25] R. J. Schalkoff, Artificial Intelligence: and Engineering Approach, McGraw-Hill, NY, 1990. [26] M. Sugeno, "An introductory survey of fuzzy control," Inform. Sci., vol 36, 1985, 59-83. [27] M. Sugeno, and T. Takagi, Multi-dimensional fuzzy reasoning. Fuzzy Sets and Systems 9(3), 313-326, 1983. [28] R. M. Tong, A control engineering review of fuzzy systems, Automatica 13, 1877, 558-569. [29] Siegfried Weber, A modification of Lukasiewicz logic and its applications to fuzziness and distances, in Approximate Reasoning in Expert Systems (ed. be M. M. Gupta, A. Kandel, W. Bandler, J. B. Kiszka), North-Holland, Amsterdam, 1985,123137. [30] L. A. Zadeh, Fuzzy sets. Information & Control, Vol. 8, 338-353, 1965. [31] L. A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. Systems, Man, & Cybernetics, vol. 3, no. 1, 28-44, 1973. [32] M-C. Zhou, and M. C. Leu, Petri net modeling of a flexible assembly station for printed circuit boards, Proc. 1991 IEEE Int'l Conf. Robotics and Autom., Sacramento, 2530-2535.

CHAPTER 20

FUZZY LOGIC-BASED TOOLS FOR THE ACQUISITION AND REPRESENTATION OF KNOWLEDGE IN BIOMEDICAL APPLICATIONS Elisabeta Binaghi IFCTR-CNR Via Ampere 56 Milano, Italy Abstract This paper illustrates the use of fuzzy logic-based tools, including a hybrid system based on neural networks and fuzzy set representation techniques, in building medical expert systems. These tools have been employed in several medical diagnostic situations presenting different complexities. The first set of applications concerns diagnostic problems in the field of gynecology; the second includes biomedical image interpretations using radiological and colposcopic images.

1. Introduction Knowledge-based techniques are widely used in the design of computer-based medical diagnostic systems to model the complex strategies and decision-making rules involved in the solution of diagnostic problems [1]. Unfortunately, these systems present limitadons when the knowledge to be represented is of any real complexity. The problem lies principally in the difficulty of defining new knowledge bases to deal with different diagnostic problems (this usually involves lengthy consultation with experts) and in that of defining a knowledge representation formalism which can provide for human flexibility and variability in formulating diagnostic judgements while ensuring a rigorous, quantitative interpretation of the knowledge underlying the diagnostic process concerned. In biomedical image interpretation and, in medical diagnosis in general, domain knowledge is strongly structured and the assignment of medical significance to signs is based on complex and obscure criteria which may be difficult to assess direcdy and transform into a suitable machine representation. An invariant characteristic of the medical domain is the intrinsic, unavoidable uncertainty arising from the complexity, multiplicity and vagueness of the concepts involved, even when these concepts are rigorously applied in a rational scheme, and expressed in a language generally accepted by the medical community. 529 S.G. Tzafestas and A.N, Venetsanopoulos (eds,) Fuzzy Reasoning in Information, Decision and Control Systems, 529-551. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.

530

Fuzzy set techniques are widely used in the solution of various practical problems in medicine: the current literature provides various examples of medical expert systems with fuzzy inference models and of medical applications using fuzzy expert system shells [2]. A major concern of these systems has been the development of fuzzy logic based methods for handling uncertainty which are both theoretically sound and computationally tractable. However the solutions adopted by a medical expert system will be of little value, unless it can guarantee the reliable acquisition of expert information. A knowledge acquisition strategy must be provided which can actively support systems by translating medical knowledge into a machine readable expression. The neural network approach has received renewed attention in the last few years. This approach offers the advantage that, unlike the rule-based approach which relies on expert input, the knowledge base is derived directly from accumulated data [3]. But, as in medicine both of these source of information are extremely important, it seems a good idea to incorporate both approaches into a single comprehensive framework. Cohen and Hudson propose the integration of neural network techniques with approximate reasoning in a knowledge system for the treatment of carcinoma of the lung [4]. Machado and Rocha present a fuzzy connectionist expert system for diagnosing renal syndromes [5]. This paper illustrates the use of fuzzy logic-based tools, including a hybrid system based on neural networks and fuzzy set representation techniques, in building medical expert systems. These tools have been employed in several medical diagnostic situations presenting different complexities. The first set of applications concerns diagnostic problems in the field of gynecology; the second includes biomedical image interpretations using radiological and colposcopic images.

2. Fuzzy set representation of medical diagnostic reasoning Medical diagnostic reasoning consists in the deduction of conclusions regarding a given case, the premises of which are the signs the patient presents and the diagnostic rules in the knowledge base. Diagnostic rules describe how combinations of clinical signs (information drawn from the patient's history and from clinical findings) relate to diagnostic classes. In diagnosis, considered as a cognitive process, the description of clinical signs and subsequent decision activity are performed not on the basis of a mass of numbers but by "elicitation Unks" between linguistic descriptions of clinical signs and qualitative diagnostic judgements. Medical experts do not use formal languages to represent knowledge and reasoning with it. The formal basis lies in deeper theory, such as physics, and this complex knowledge may be subject to imprecision and vagueness. However, physicians have developed fairly stable procedures and a generally accepted language for representing knowledge and reasoning with it. Any effective solution to the acquisition and representation of knowledge in medical domains must take into consideration the following aspects:

531

•

the acquisition of medical diagnostic knowledge is inherentiy biasing. A strategy with a relaxed structure may facilitate data elicitation and improve the quality of the acquired knowledge; • success in representing uncertain knowledge depends on the compatibility of the language used with both the mental representation of concepts in the human mind and the computational constraints of machine representation. An approach that deals with all these aspects consists in combining the use of fuzzy sets as the representational framework with a machine-aided knowledge acquisition strategy which includes fuzzy learning techniques, neural networks and structured interviews for the generation of fuzzy knowledge. Fuzzy sets, as proposed by Zadeh [6], address the representation problem by providing adequate knowledge structures when the information is uncertain due to vagueness and imprecision [7]. Machine learning and neural networks can faciUtate the process of knowledge acquisition by transforming readily available knowledge, such as examples, into a useful form and by reducing ill defined interactions between expert and engineer. There are several potential alternatives for the acquisition and structuring of fuzzy knowledge in medical expert systems. The choice depends on the characteristics of the domain, on the problem-solving method used by the experts involved, and on the previous experience and perception of the knowledge engineering. The fuzzy set-based approach to knowledge representation has been amply investigated and has its roots in a seminal paper by Zadeh [8]. The general methodology proposed there, is descriptive rather than normative and several specific questions must still be solved in order to transform the methodology into an effective strategy. We have defined several solutions for acquiring and structiuing fuzzy knowledge and inference models in medical diagnosis, drawing on the strenghts of the most suitable approach for each situation. The solutions adopted have been implemented with a general and flexible set of tools addressing the following main tasks: • membership function elicitation techniques • definition of fuzzy knowledge structures, including: • fuzzy logic to represent declarative propositions • fuzzy frames describing hierarchies • fuzzy production rules for causal knowledge representation • elicitation and generation of diagnostic decision rules

3. Fuzzy declarative propositions for the description of clinical signs in gynaecological problems The fuzzy representation model interprets qualitative concepts involved in the diagnostic problem as fuzzy sets. In particular, a clinical sign may be interpreted as a linguistic variable and its description represented as a fuzzy declarative proposition of the form X is A, where X is the linguistic variable representing a given sign and A is a term belonging to the term set of the linguistic variable and denotes a fuzzy set in a given universe of discourse f/ = (M,,....,M„) characterized by the membership function |i^(M). The set U contains all the possible numerical or, more generally, crisp values assumed

532

by X. The fuzzy set denoted by A defines a possibility distribution of X . In this context, the definition of a strategy to acquire and interpret expert data to characterize fuzzy declarative propositions concerned with the problem under investigation, is a crucial step, which may drastically affect the performance of the knowledge representation process in general. A relaxed strategy in acquiring knowledge is essential here to preserve the variability of expert judgements, but at the same time an objective procedure which can quantify and aggregate expert answers must be introduced. The elicitation strategy must take into account the fact that due to the nature of fuzziness, people will not evaluate a subjective attribute identically and that different answers will be collected from different assessors. It is important to select an appropriate group of experts selected for the elicitation session in the sense that they must hold welldefined opinions on element membership in the fuzzy sets in question and shared concepts. Experts are therefore considered fuzzily consistent if a 6 exists depending on the experiment in hand, such that |J.^,(x^O-ii^,(Xy) < 8 , in which |x^, , \i.^,

are

membership functions for fuzzy sets f] as perceived by a and b , in which a , b is any subject of the 'group' [9]. 3.1.Membership function elicitation techniques A fundamental task in acquiring fuzzy knowledge is the defmition of representative membership functions for the fuzzy sets concerned. Analytical formulations of the distribution of possibility over a given numerical range must be provided, but these are difficult to obtain directly from experts. In our approach the construction of membership functions proceeds from the selection of an appropriate group and is based on a statistical surve^. Different approaches have been investigated to construct membership functions using statistical data and constrained optimization techniques are indicated to find the optimal membership function [lOJ. Our method is to ask the group of experts for a scaled response. Once N responses concerning the degree of confidence have been obtained, the representative response is found by averaging the N responses and applying fitting operations on the resulting rough distribution. This may seem to contradict the inherently individual nature of uncertainty, but, if the focus of attention is not individual's evaluation of a phenomenon, but the phenomenon itself, the approach is justified. 3.1.1.Representation of clinical signs in the domain of postmenopausal osteoporosis In some cases experts could refer the distribution of possibility associated with a given fuzzy label to a standard function and the system then elicits membership functions by adjusting suitable parameters in standard piece wise functions [11]. The parameters are cardinality measures of intuitive information about the distribution of possibility and may be elicited directly from the experts without the risk of creating artificial information. This above technique has been amply applied to build a forwardchaining fuzzy rulebased system in the domain of postmenopausal osteoporosis [11]. The application

533

has had to cope with the huge number of clinical signs involved and the lack of standardization in defining borderline conditions. A l clinical signs and diagnostic classes have been encoded as linguistic variables and their descriptions fo^rmalized in trams of fuzzy declarative propositions. Diagnostic rules have been formalized and stored in terms of a set of fuzzy conditional statements. The elements considered were 29, including hematic and chemical parameters, mineralometric parameters, generic risk factors aad specific risk factors. Most of the clinical signs involved in the diagnosis had terms represented by standard functions. Figure 1 shows the standard membership functions of the linguistic tenms used by physicians to characterize the Bone Mineral Content of the Radius.

Figure 1 -Membership functions for tlie clinical sign bone mineral - content Tliese functions result from an elicitation process, in which preliminary data are acquired (in the example in figure 1, the numerical range and the type of instrument with which the numerical values are obtained). For each function, the experts partition the numerical range, identifying the intervals in which they judge the compatibility of numerical values with the linguistic label concerned high, medium or low. Parameters for the specification of standard piecewisc functions are computed directly from this information, taking the values at the extremes of the intervals estabhshed by the experts, and applying the "mean" operator to average the different values obtained.

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This procedure has been applied to elicit membership functions for 62 terms. The physicians interacted directly with an intelligent interface, providing data and validating results. A fuzzy matching procedure has then applied for a quantitative evaluation of the membership functions elicited. The degrees of compatibility between the numerical values of clinical signs and the linguistic terms were collected firom the experts, the corresponding degrees of membership computed using the analytical formulation of membership functions. The comparison took into account the intrinsic variability in experts' answers. Exact matching would not provide a reliable evaluation of the agreement between experts and automated procedure. The problem has been addressed by estimating, for each degree of similarity resulting from the implemented membership functions, a range of acceptance R = [\xia)-t,[i(a) + t] where / is a value related to the variabiUty in experts' answers [12]. They were assessed to agree if the human degree of compatibility lies within the corresponding R A test set of 700 samples from the universes of discourse of all the memberships (62 terms) showed an agreement of 82%. Disagreement was confined to membership functions representing borderhne conditions. 3.I.2.Representation of clinical signs in the clinical assessment of body hair growth in women There are complex situations in which it is difficult to predict the trend of the membership functions. In these cases data elicited from experts serve to determine not only representative values for parameters of known functions, but the overall distribution of an unknown trend as well. This technique has been used in a medical application aimed at formaUzing and automating the clinical assessment of body hair growth in women for use of hormonal disorders. This chnical problem has usually been approached as if there were a clear dividing line between the normal and the abnormal state and assessment has been based on a quantitative, crisp method providing a numerical score to represent the clinical evaluation [13]. The resulting discrepancies among experts and unpredictable subjectivity in the diagnosis have suggested that a fuzzy medical expert system could better guarantee an objective, expUcit and deterministic assessment of the diagnostic process. The clinical signs involved in the evaluation of hair growth are density, consistency and area. The difficulties encountered in creating fuzzy sets for these signs are due principally to the global visual perception underlying the clinical evaluation.. Visual data, in the form of scenes and pictures, are often processed in visual terms alone, without any corresponding translation or recording as verbal labels or representations [14]. This was a crucial aspect in the elicitation of fuzzy sets for the area sign. Experts were unable to qualify linguistically it and required of contextual information to provide reliable evaluations. We addressed the problem by implementing a graphical interface displaying all the elements of the body as parts of an integrated whole, and not as separate and independent items. Figure 2 shows the interface window for the acquisition of membership functions related to the sign area evaluated in the chin site.

535 firade

nwr m J.

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2

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NO

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NO

5

NO

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Figure 2 - Elicitation of the membership fanctions of thesign area The global region of ioterest was divided into 5 items. Progressive combinations of these items were displayed: they constituted the range on which the membersMp functions for the linguistic terms significant, borderline, unsignificant are defined. Because it was impossible to procure explicit descriptions of the sign area, these terms were semantically related to a meta^diagnosis in which, together with the area, consistency and density parameters are evaluated by the experts. Following the membership function exemplification method [15], the experts were asked to express the compatibility of each term with each combination of items by answering yes or no and assigning a number from 0 to 5 to kidicatc their degree of confidence in the answer. The collected data were then processed to produce the representative value |i,(x)of the membership function for the term t and the combination of area items x.. The following formula were used:

\i,{x) = - + dx(—) 2 10 where d-l if the answer was yes and d-Q if the answer was no. The degrees n,(x), quantifying the compatibihty between the linguistic term t and the crisp value assumed by the cliflical sign concerned for each expert, were aggregated. Several measures were

536

used to group the different answers; among these were the mean, median, expected value and the fuzzy expected value. Of all these, only the fuzzy expected value seemed to give consistent results: extreme values did not greatly affect it, as they did the other measures, and, despite the great variability in answers, it provided a measure indicative of some sort of a central tendency [16], Figure 3 shows membeiship functions for the sign area obtained by applying a splinebased fitting operation on the representative values computed with the fuzzy expected value. The fuzzy matching procedure was applied to compare membership functions values and expert judgements. Global agreement, of 75%, evaluated on a training set of 50 sample values, was found. A partial agreement of 90% was found for clinical signs for which the fuzzy consistency was fully respected.

Figure 3 - Membership function of fuzzy sets for the sign area 4. Fuzzy Frames for Highly-Structured Descriptive Knowledge Knowledge-based systems for biomedical appMcations are often called upon to formalize and store strongly structured descriptive knowledge regarding medical descriptions of the anatomical parts concerned, feature descriptions and static relations among objects. In particular, when dealing with biomedical image interpretations, the underlying knowledge is characterized by a large number of items and complex hierarchical relations among them.

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a-priori anatomical knowledge about intrinsic properties of human organs and imageprocessing knowledge about properties of the image involved in the application. The fuzzy frames are used as a generalization of the knowledge structures introduced in AI with the aim of representing in a unified framework multi-typed objects characterized by multiple uncertainty. The frames represent objects in terms of a set of attributes which may be value-attributes, specifying by means of assignment or procedure computation the value or range of values the attribute of a given object should have, or relational attributes denoting links with other objects. In real applications, where attributes are very often fuzzy in nature, independent levels of uncertainty may be integrated in the frame structure to represent the degree of certainty with which a value is assigned to a given object attribute, or to define gradual multivalued predicate links among objects. Several attempts to formalize the concept of fuzzy frames are described in the literature [17]. We have focused our attention on fuzzy frame representation for biomedical image recognition purposes, introducing two kinds of frame: model frames, representing models of anatomical parts and object frames, representing the description of specific objects in the current image and created when an object is recognized as an istance of a given anatomical model. Frames representing models are predefmed on the basis of the knowledge provided by the experts and describe classes of objects. In each model-frame, own-slots list characterizing attributes of the class, and member slots describe those attributes that characteriaze an element of the class and assume a significant role in the development of the interpretation process of the final image. Object frames are instances of model frames and contain a member-of link to the class to which the object represented by the frames belongs. Slots of the object frames are own slots and represent the attributes of the object; these slots are coincident with the member-slots of the corresponding model frame. In order to deal with situations in which knowledge related to a given object is qualitatively or intrinsically imprecise, fuzzy slots are introduced in model and object frames, and a fuzzy inheritance mechanism is defined to represent partial belongingness of an object to a class represented by a model frame. 4.1 The use of fuzzy frames for the representation of knowledge in the analysis of CT images of the vertebrae To assist the presentation of the methodology with an explanatory example, we refer to the problem of the analysis of CT images of the vertebrae (figure 4). The expected result of the analysis is a quantitative evaluation of the bone inineral content, related to metabolic pathologies of the skeleton, such as osteoporosis. The analyst's procedure consists in identifying the spongy bone, outlining the Region of Interest (ROI), excluding the area of high vascularity, and computing the bone mineral content in the ROI on the basis of conversion factors obtained from analysis of calibration phantom regions. In this application context, the knowledge base contains model frames representing anatomical parts such as lombar vertebra, cortical bone, spongy region.

Figure 4 - A multiwindow interface for the recognition of CT images of vertebrae Object frames are related to those objects in the current images that are recognized or created in accordance with the goals set. In particular, object frames for cortical bone and spongy region are created as a result of the recognition procedure. An example of the model and object frames involved in the analysis of CT images of Vertebrae is given in table 1. The o&jert/rame SP-RE is a fuzzy frame in two respects. First, the degree of property inheritance from the model frame in the member-of slot may be specified as a number between [0,1] (in the case of no values specified, the default value 1 is assumed). Second, the other slots may contain fuzzy sets as values. The degree of membership specified in the member-of slot documents the results of the fuzzy reasoning process underlying the frame structure and dedicated to the recognition task. The reasoning activity for recognizing as Spongy Region a candidate Obj-i , resulting from a previous segmentation procedure, is based on the evaluation of the degree of matching between attributes in own slots of model frame of Spongy Region and attributes of the candidate object. The matching evaluation impMes the execution of procedures contained in the own slots of Spongy Region model frames. The procedures Below and Surrounded require that object frames for Spinal Canal and Cortical Bone respectively be already recognized. A metarule in the Knowledge-Base subordinates the activation of the recognition rule to this condition. Spinal Canal is recognized by a thresholding procedure, and Cortical Bone is recognized as the region

539

expresses the discriminant condition that the shape of the spongy region is circular, conceiving as circular a fuzzy concept with an intrinsic variability in its definition.

Model Frame : Spongy Region Part of: Lombar Vertebra; Own-slot: Position-below Own-slot : Position-surrounded Fuzzy Own-Slot: Shape Circulr Fuzzy Own-Slot: Shape-Ray Member Slot: Density Member-Slot: Position of Centroid Fuzzy Member Slot: Homogeneity

Object Frame : SP-REl Member of : Spongy region [1] Own-Slot: Density Own-Slot: Position of Centroid Fuzzy Own-Slot "Homogeneity

Table 1- Structure of Frames The evaluation of the degree of circularity of a given object is made by applying a measure of fuzzy entropy to the accumulation matrices produced by Hough transform procedure. Entropy provides a measure of the degree of "spreading" in accumulation matrices and then a measure of the degree of circularity. Entropy values are the universe of discourse of a fuzzy set whose inembership function provides the degree of circularity of a given object in correspondence with its entropy value. The resulting advantage of having represented the recognition procedure in a frame structure and in underlying fuzzy recognition rules is that the overall knowledge has been explicitly represented and its role synthetically codified. Partial belongingness of an element to a real-world class is quantified in fuzzy member-of slots. The degree of property inheritance results from the recognition activity which is based on the evaluation of fuzzy and/or nonfuzzy attributes. This application has successfiiUy addressed the problem of accelerating the quantitative analysis of vertebral bone mineral content and of providing more objective procedures for reaching a consensus among experts. A complete evaluation of the results may be found in [18].

5. Machine learning techniques for knowledge acquisition in fuzzy environments Difficulties may arise in ehciting the overall set of diagnostic rules directiy from the experts of a given medical domain. The interview technique, although it may be more successful than other methods in revealing the detailed structure of a concept, presents the risk that the knowledge elicited may be an artificial creation produced under the pressure of questioning. Leaming-by-example strategies offer domain experts tools for the direct transfer of their specific, exemplified diagnostic knowledge and provide inductive algorithms to induce knowledge in a general abstract form. Various methods for generating fuzzy production rules or, fuzzy relations in general, have been proposed in the

540

literature [19]. Our approach focuses on an empirical learning strategy which provides a reasonable explanation and monitoring of intermediate results and is consistent with the possibilistic, rather than probabilistic, nature of uncertainty [20]. 5.1.Fuzzy logic-based classification of cendcography images An empirical learning procedure has been applied to the problem of designing an automatic image interpretation system to support the classification of colposcopic images for the identification of neoplastic and preneoplastic regions. This could be a valuable contribution to the generation of diagnostic rules in a sector where the intrinsic complexity of the scene and the large amount of information the physician must correlate in his visual analysis make interview techniques inappropriate. Cervical intraepithelial neoplasia and small invasive cancer of the cervix are in most cases asymptomatic diseases and consequentiy active screening and early detection are essential. Recent studies advise integrating the traditional Papanicolaou Smear with visual inspection (colposcopy) [21], But colposcopy, in spite of its amply proved capability of reducing the percentage of wrong diagnoses ("false negatives"), is not widely employed, basically because it is both expensive and time consuming. Cervicography is a recent technique introduced as a new screening model for colposcopic examination. The basic idea of cervicography was to develop an optical instrument, the cerviscope with which any health care person, even without any knowledge of colposcopy, can take a standardized picture of the cervix and send it to an expert for evaluation. The possibility of defining standardization protocols and employing cervicography in large population groups suggested the application of automatic image interpretation techniques to digitized cervicography pictures in order to identify patients at risk for neoplasia of the cervix. In this situation, an automatic system supporting medical diagnosis, by selecting from a large number of subjects those showing some suspect alteradon of uterine epithelial tissues, could substandally reduce (by about 60%) the population group for which further examination is recommended. The first important goal is to be able to classify subjects in groups identified on the basis of characteristic features of an epithelial region called the "transformation zone", where the anomalous alterations appear in the early stage of the desease. The diagnostic group ranges from subjects for which no further investigation is currendy required to subjects for which visual inspection or additional colposcopically directed biopsy is recommended. 5.2. The image model In the cervicography images, the scene observed is the uterine cervix with its different types of epithelial tissues. The region of interest is the squamocolumnar junction, that is, the transformation zone in which possible alteration processes are located. This is the locus where metaplastic or neoplastic lesions generally appear. Figure 5 presents details of cervocographic images showing the region of interest of several subjects.

541

Figure 5 - Eegions of interest in cervicographic images In his visual inspection , the expert bases his judgement on botli variations in the color of tlie different tissues, and on featurc-s of the sqoamocoliimnar junction . Several types of lesions can be differentiated by the expert. We restrict our analysis to three classes, to reduce the complexity and in view of the fact that even this first, simplified level of classification may have a significant and immediate impact in the screening activity. Tlie classes identified are: • ECTOPIA, grouping normal subjects, which present under colposcopy only normal malpighian and colnmnar epithelia and no abnormal lesions. The junction zone has a sharp and imegular boundary and the color varies from the internal to the external region, from dark to light. • NTZ, where an immature squamus metaplasia is present. The junction is fuzzy and dark to light. • ATZ, identifying subjects for which colposcopically diiBcted biopsy is prescribed. The junction is well defined and regular and the color difference is from light to dark. Classification is achieved with a complex interpretation strategy which consists mainly in selecting and performing those ¥isual procedures which first focus analysis on significant regions in the scene and subsequently allow the extraction of discriminant features. For the sake of brevity, only classification tasks are described here. A

542

detailed description of the preprocessing and processing activities involved in the application may be found in [22]. The classification procedure involves the evaluation of the selected features which characterize the structure of interest, and the decision to assign these structures to a given diagnostic class. Wide and intrinsically uncertain knowledge underlies the visual inspection of the scene. The huinan decision-making activity of assigning medical meanings to image patterns is based on implicit criteria which are difficult to assess and transform into a suitable machine representation. We have addressed these problems by designing the classification procedure as a fuzzy rule-based system and by using automatic learning techniques to generate fiizzy classificatyion rules [ 11,22]. The classification rules are encoded in fuzzy production rules structured as evaluationdecision pairs . They define a fuzzy relation between the qualitative description of features and decision classes. The strength of implication in fuzzy production rules is approximated by a linguistic variable reflecting the intrinsic human variability and vagueness with which a conclusion may be inferred. Difficulties may arise in eliciting fuzzy production rules direcdy from experts and, in the expression of the different strengths of implication, which may vary from rule to rule in particular. We propose an intuitive learning by examples technique which uses fuzzy reasoning operators in the form of fuzzy induction to automatically infer the strength of implication, given a training set of already classified images. In this application, classes were assigned on the basis on the selected features: sharpness, morphology, color tranmsition . In the fuzzy classification scheme each feature is considered a linguistic variable with the following associated term sets: • Sharpness = ( well defined, medium, fuzzy) • Morphology = ( regular, medium, irregular) • Color Transition = ( light-dark, dark-light) Terms in the above term sets are fuzzy sets having as their universe of discourse the numerical ranges of the corresponding features , provided by the described above measuring procedures. The classification rules were modeled in terms of fuzzy production rules having the following general form [8]:

/4i is a fuzzy compound declarative proposition aggregating linguistic descriptions of the featiu^es sharpness, morphology and color transition. D^ belongs to the set of diagnostic classes D= ( Ectopia, NTZ, ATZ). w^ belongs to the term set of the linguistic variable W, the values of which are the possible linguistic expressions of the strength of implication. Terms in the term set are fuzzy sets defined in the interval [0,1]. We reduce them for the sake of simplicity to a fuzzy number by applying a suitable defuzzification algorithm.

543

The problem of learning fuzzy classification rules is formulated in terms of how to automatically induce , from a training set of examples T* the strength of implication H;* for each rule /?, and for a given diagnostic class D*. These examples are classified image structures , each of which may be formalized in the following form:

t'=lu,J^]u'eT' where u, is the vector of measurements obtained in correspondence with the selected set of features, ;* is the expert's judgement expressing the strength or degree of satisfaction with which the image structure represented by u, may be assigned to class The experts use predefined linguistic labels, such as very high, medium, low, very low to express their mental judgements; these labels are automatically translated into numbers in the interval [0,1]. Given the diagnostic class D^, let: • /, be the cardinality of the set of all the possible antecedents ( -4, ) • m, the cardinality of the training set T^ • p, the cardinality of the term set W containing the terms with which the strength of implication may be expressed

Matrices A/* and Mj are then defined in the following form:

\^A>^)

^i.,(«J M,

A/r = H^,(«l)

\^A,(UJ

^i^.0'^)

^.,0•»)

Using max-min operators, we perform the matrix product: G = A/*oM* to obtain matrix G with dimension Ixm. The element g^ expresses the induced degree of certainty with which the corresponding rule may be generated. This learning procedure has been applied to the classifcation of cervicographic images using a training set of 120 images. Twenty-six classification rules have been generated. When these were applied on a test set of 50 subjects, they gave an accuracy of 79%. Table 2 shows a mle describing a typical case of ATZ. Figure 6 shows the main phases of the overall image interpretation procedure, including segmentation, contour detection and classification, applied to three different subjects.

544

'colposcopic

6. A symbiotic fuziy neural approach for the acquisition and representation of medical knowledge Current trends in designing inteligent systems focus on integrating fuzzy logic-based (FL) tools and neural network (NN) techniques, to combine the suitability and flexibility of fuzzy logic in representing cognitive processes with the effectiveness of neural networks in learning knowledge from examples. The symbiotic relationship between the two computational paradigms has attracted the attendon of many researchers in the last few years who integrate FL tools in connectionist models, or NN techniques in fuzzy logic frameworks [3J. These hybrid systems can support different levels of interaction where NNs and FL are used in cooperation on the same task. A unified framework supporting both low and high interaction between the two models is presented here and applied to the acquisition and representation of medical diagnostic knowledge in the domain of hormonal disorders.

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The application has been introduced in section 3.1.2, detailing the strategy adopted to elicit and representing the fuzzy sets characterizing the linguistic descriptions of visual clinical sign concerned. In this application, a fuzzy neural franiework has been designed and implemented to model diagnostic relationships between clinical signs and diagnostic judgements. 6.1.The clinical assessment of body hair growth in women The clinical assessment of body hair growth in women used in the study of hormonal disorders, proceeds by evaluating clinical signs, such as area, density and consistency in 9 different body sites. A metadiagnosis is made for each body sites and partial diagnostic judgements are then aggregated to perform the final diagnosis. The overall diagnostic process has been implemented by an hybrid intelligent system which uses fuzzy trained NN and fuzzy production rules in a unified framework. 6.2. The neural stage The metadiagnostic reasoning designed aimed to produce a diagnostic judgement for a given body site has been implemented with a connectionist model. Knowledge underlying the diagnostic stage is complex, difficult to assess, and difficult to represent in symbolic form, due partially to the global visual perception underlying the clinical evaluation of each body site. Neural network (NN) techniques can successfully address these problems by providing accurate learning mechnisms capable of generating and/or modifying knowledge structures autonomously. In particular NNs offer the means to overcome some of the major drawbacks of conventional knowledge acquisition techniques, such as the lengthy consultation with experts required in structured interview techniques, and the design of complex conceptual models when symbolic machine learning algorithms are adopted. NN learning mechanisms mantain their ability to learn in uncertain and unknown environments. We propose for building NNs into the hybrid framework for medical diagnostic reasoning the use of a feed forward one hidden layer neural network with an adaptive, problem-dependent back-propagation learning algorithm. A detailed description of the NN model and of the learning mechanism adopted can be found in the references Binaghi te al. [23]. The reasoning activity involved in the evaluation of body hair growth has been modeled in any single given body site as a classification process which can assign an event, i.e. a set of linguistically qualified clinical signs (for example area is significant, density is sparse, consisteny is terminal), to one or more of several predefined categories. The implemented NN classifier must compute the degree of similarity (membership of possibility) of the event being analyzed to each one of the prototypal hypotheses defined in the model.

.•546

Thus each sub-diagnosis (of a body part) is characterized by a set of input events, a set of output hypotheses, and a NN implementing a fuzzy relationship between input and output. The NN assumes a feed-forward topology with three layers. Each input node represents the value of an attribute of the object (or event) to be classified; in our approach each term linguistically qualifying a clinical sign is represented by a different neuron . The degree of confidence in each of these terms (obtained by evaluating its membership) is represented by the state of activation of the neuron. Hidden neurons represent intermediate abstractions. The output layer represents the hypothesis. In particular the state of activation of each output neuron represents the degree of possibility of the event under analysis to the hypothesis represented by the neuron. In our application there is one diagnostic class, corresponding to the clinical assessement of body hair growth in women; the different hypotheses are linguistically qualified degrees of severity, ranging from very low to very high. The sigmoid function is used to determine the output state. The output value V, from the i-th node is: V.=-

1 l + e '"

where m is the number of nodes contributing and V. is the output value of the j-th node. 6.3.Problem dependent learning rule in feed-forward neural networks Back-propagation, which is also known as the generalized delta rule, is one of the most popular and widely investigated learning methods used to train feddforward neural networks. The back-propagation learning algorithm is based on nonlinear optimization technique of gradient descent on the sum of the squared differences between the activation of the nodes in the output layer and the desired output. Application of the gradient descent method yields iterative weight update rules [24]. The application of neural network learning mechanisms presents some limitations, despite their ability to leam in an uncertain or unknown environment. A comparison of experimental results in classification application shows advantages, but also disadvantages, with respect to other methods employing knowledge-based approaches [25]. The main drawback of the NN method lies in the fact that the generalized delta rule is computationally complex: when the training data are numerous the learning time can be very long. Moreover, the generalized delta rule can not guarentee that the global error minimum will be found, and the solution strategy depends on the initial network configuration (often assigned at random).

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Several attempts have been made to improve the performance of back propagation algorithms; fast learning algorithms heve been proposed to accelerate the training phase [26]. Our approach integrates problem-dependent knowledge in the analytical formulation of the back propagation algorithm in order to control and subsequently improve the weight update procedure. TTie central concept is that of subordinating the activation level of one or more hidden neurons to the presentation of certain input pattern configurations to input layer. The fuzzy logic framework is used to support the representation of this knowledge, which may be uncertain and/or linguistically qualified. The flexibility of the fuzzy reasoning approach makes it easy to define and update rules until a reasonable performance Ls achieved, and to introduce new intuitive knowledge at different levels of abstraction in a natural and controlled way. Fuzzy production rules, structured as evaluation-decision pairs, are used as knoweldge structures to represent all the ingredients of the fuzzy reasoning process. In the antecedent of the rules, conditions related to input patterns are expressed in terms of compound fuzzy declarative proposistions. The decision part of the rules contains fuzzy declarative propositions which describe linguistically the values that variables related to hidden neuron activation assume when the corresponding evaluation part of the rules is satisfied. We have defined a forward chaining fuzzy inference mechanism to interpret these rules: it assigns a degree of confidence to each rule in the knowledge base, using a fuzzy matching procedure, and deduces the best hidden neuron activation through a defuzzification process applied to the variables in the consequent part of rules. We use the fuzzy logic based learning algorithm to train the networks devoted to each diagnostic subproblem involved in the medical application concerned. 6.4.Neural network implementation At the present stage of our experiments, preliminary versions of the NN classifiers have been implemented for each of the 11 body sites examined. Figure 7 shows the topology of the neural network used. The number of input nodes is determined by the dimension of the input patterns, which are degrees of evidence specified for each term of the clinical signs. Input neuron activation lies to the range [0,1]. Each output neuron represents a degree of possibility of assigning the current event to a certain degree of severity and also has activations in the range [0,1].

548 veiylow

low

medium

high

vciyhigfa

o o density

consistency

Figure 7 - Neural network topology Each of the NNs implemented have been trained using training sets constituted by diagnosed clinical cases. A total number of 400 examples have been considered. The fuzzy logic-based NN has been implemented using different knowledge bases in function of the site. For the chin for example, 1 hidden neuron has been constrained to the presentation to the input layer of evidence confuming a low degree of severity. Table 2 lists a subset of rules in the knowledge base for the chin site. • • •

1/ density.sparse is excited and consistency.vellum is excited Then hidden neuron-1 is excited If density.sparse isrelaxedand consistency.vellum is excited Then hidden neuron-1 is relatively relaxed / / density.sparse is relaxed and consistency .vellum is relaxed Then hidden neuron-1 is relaxed Table 2 - Rules in the knowledge base

Suitable membership functions have been elicited to characterize linguistic terms in the rules.

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6.5.Preliminary results To assess the diagnostic accuracy of the implemented neural networks and their classification capability, a test set of examples, consisting of already diagnosed clinical cases referred to specific body sites, has been used. The testing data considered were 200. The diagnosis was confirmed in 82% of the cases. 6.6.Work in progress We are currently engaged in experimental work to conclude the application and the model the overall diagnostic process. The cognitive problem underlying this application has been broken down into a hierarchy of classification tasks: • first level: diagnoses applied to the body sites • second level: aggregation of first level classification results to produce a final judgement on total body hair growth. We are working on the second level task at present, experimenting procedures to aggregate partial results: • •

a procedure using an higher level NN; in this case the outputs of the first level are the input for the higer NN; a procedure using a fuzzy rule-based system; in this case the symbolic learning procedure presented in previous section is adopted to the problem under investigation.

The two solutions present unique advantages and disadvantages experimentation has been designed to exploit and evaluate each.

and

further

References 1. Szolovits P., Patil R.S., Schwartz W.B., Artificial Intelligence in medical diagnosis. Annals of internal medicine, 108, pp.80-87, (1988). 2. Adlassnig K.P., Kolarz G., SheithauerW., Present State of the Medical Expert System Cadiag-2, Meth. Inform. Med., 24, 1, pp.13-20, (1985). 3. Kandel A., Langholz G. (Eds.),"Hybrid Architectures for Intelligent Systems", CRC Press, Boca Raton (Florida), 1992. 4. Hudson D.L., Cohen M.E., Banda P.W., Blois M.S., Medical Diagnosis and Treatment Plans Derived from a Hybrid Expert System. In "Hybrid Architecture for Intelligent Systems", (Kandel A., Langholz G. Eds.), CRC Press, Boca Raton (Florida), 1992.

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5. Machado R.J., Rocha A.F., A hybrid Architecture for a Fuzzy Connectionist Expert System. In "Hybrid Architecture for Intelligent Systems", (Kandel A., Langholz G. Eds.), CRC Press, Boca Raton (Florida), 1992. 6. Zadeh L.A., Fuzzy Sets, Information and Control, 8, pp.338-353, 1965. 7. Zadeh L.A., The concept of Linguistic variable and its Application to Approaximate Reasoning. In "Fuzzy Sets and Applications", (Yager R.R., Ovchinnikov S., Tong R.M., Nguyen H.T. Eds.), pp. 193-329. John Wiley & Sons, 1987. 8. Zadeh L.A., PRUF - a Meaning Representation Language for natural Languages. In "Fuzzy Reasoning and its Applications", (Mamdani E.H. and Gaines B.R. Eds.), Academic Press, 1981. 9. Hall L., Szabo S., Kandel A., On the Derivation of Memberships for Fuzzy Sets in Expert Systems, Information Science, 40, pp.39-52, 1986. 10. Civanlar M.R. and Trussel H.J., "Constructing Membership Functions using Statistical Data", Fuzzy Sets and Systems, 18, pp 1-13, 1986. 11. Binaghi E., A fuzzy Logic Inference Model for a Rule-based System in Medical Diagnosis, Expert System, 7, 3, pp. 134-141, (1990). 12. Binaghi E., Delia Ventura A., Rampini A., Schettini R., A fuzzy Reasoning Approach to Similarity Evaluation in Image Analysis, Int. J. of Intelligent Systems, 1993, in press. 13. Ferriman D.M., Gallwey J.D., Clinical Assessment of Body hair Growth in Women, J. Clin. Endocr. Metab., 21, 1440-1447, 1961. 14. Freedman J., Haber R.N., One Reason Why We Rarely Forget a Face, Bull. Psychon. Soc.,3,pp.l07-109, 1974. 15. Chameau J., Santamarina J., Membership Functions I: Comparing Methods of measurements, //;/. J. of Approaximate Reasoning, pp.288-317, 1987. 16. Pal S.K., Fuzzy Tools for the Management of Uncertainty in Pattern Recognition, Image Analysis Vision and Expert Systems, Int.J. System Science, 22, 3, pp.511549, 1991. 17. Yager R. R., Linguistic representation of default values in Frames, IEEE Tran. on Systems Man and Cybernetics, SMC-14, No.4, 1984. 18. Binaghi E., Delia ventura A., Rampini A., Schettini R., A Fuzzy Knowledge-Based System for Biomedical Image Interpretation. In "Uncertainty in Knowledge Bases", (Bouchon-Meunier B., Yager R.R. Eds.), Springer-Verlag, Beriin, 1991. 19. Delgado M. Gonzalez A., The frequency of Fuzzy Domains and its Application to the System Identification, in Proceedings of the 2nd International Conference on Fuzzy logic and neural networks, Ilzuka (Japan), 1992. 20. Binaghi E., Empirical Learning for Fuzzy Knowledge Acquisition, in Proceedings of 2nd International Conference on Fuzzy Logic and Neural networks, Ilzuka (Japan), 1992. 21. Darnell Jones D.E., Creasman W.T., Dombroski R.A., Lentz S.S., Waeltz J.L., Evaluation of the atypical Pap Smear, Am. J. Obstet. Gynecol., pp. 157-544, 1987. 22. Schettini R., Low-Level Segmentation of Complex Color Images. In "Signal Processing VI: Theories and Applications", (Vandewalle J., Boite R., Moonen M., Oosteriinck Eds.), pp.535-538, Elsevier Science Publishers, 1992.

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23. Binaghi E., Mazzetti A., Orlando R., Rampini A,, Integration of Fuzzy Reasoning Techniques in the Error Back-Propagation Learning Algorithm, in Proceedings of 6th Italian Workshop on Neural Nets, Vietri sul Mare (Salerno), 1993. 24. Rumelhart D.E., Hinton G.E., Williams R.J., Learning Internal Representation by Error Propagation. In "Parallel Distributed Processing", (Rumelhart D.E., Mc Clelland J.L. Eds.), MIT Press, Cambridge MA, 1986. 25. Binaghi E., Rampini A., Fuzzy Decision-Making in Classification of Multisource Remote Sensing Data, Optical Engineering, 1993, in press. 26. Vogl T.P., Mangis J.K., Rigler A.K., Zink W.T., Alkon D.L., Accelerating the convergence of the back-propagation method, Biological Cybernetics, 59, pp.257263, 1988.

CHAPTER 21

FUZZY LOGIC DESIGN OF A NONDESTRUCTIVE ROBOTIC FRUIT COLLECTOR S.G. Tzafestas, F.V. Hatzivasiliou^*) and S.K. Kaltsounis Intelligent Robotics and Control Unit (IRCU) Coinputer Science Division National Technical University of Athens Zografou 15773, Athens, Greece (*) Also with: Hellenic Air Force Research Center Terpsithea, Ano Glyfada, Athens 16501, Greece

1.INTRODUCTION After many years of research, the "repetitive movement robot" seems to have reached its peak of evolution and has become nowadays a routine in the product line. Recently, research was directed towards the development of machines that are able to perform more complicated tasks. Examples are robots equipped with vision, built-in intelligence, doublearms and so on [1,2]. In this paper an attempt is made to create a higher level robot equipped with vision and artificial intelligence. An interesting application of artificial intelligence in robotics can be found in the Hirota - Aral - Hachisu project [5] where a fuzzy algorithm of twenty four rules is used to estimate the distance that the gripper must traverse according to the speed and the present distance between the gripper and the object on a belt conveyor. Fuzzy logic theory was first introduced by Zadeh [4] and suggests a mathematical formulation for the linguistic sets and human reasoning. In this way the human style of reasoning can be simulated by a computer. The fuzzy logic has been successfully applied to industrial and chemical process control [6,8] and recently attempts have been made to apply its methods to the control of robot arms. In this paper, by "non destructive collection of fruits" we mean the process of collecting a fruit without destroying the actual tree. Two main issues that must be considered in any attempt to create a machine capable of non-destructive collection of fruits are the following: a) The fruits of most types of trees, and particularly the apple tree, are found in bunches. The fruits of a certain bunch are connected to some point of the tree and thus, as a result of this, each fruit of a certain bunch must be detached towards the side at which it is connected with a bunch. The robot, through the picture it receives, must determine the side on which it must be detached. b) It is possible that a fruit of the bunch may not be directly visible to the robot, as its line of sight may be blocked from another fruit of the same bunch. This must be taken under consideration. From the above it is seen that the task of non-destructive collection, demands a robot with vision (in order to recognise the fruits) as well as with artificial intelligence (in order to infer a direction in which each fruit must be grabbed and detached). The fact that a fruit must be detached from a certain side and not in a specific direction, gives an error tolerance which is very helpful as far as the final solution of the problem is concerned. The direction 553 S.G. Tzafestas and A./^. Venetsanopoulos (eds.) Fuzzy Reasoning in Informalion, Decision and Control Systems, 553-561. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.

554

of approach and detachment of each fruit is inferred through a fuzzy algorithm giving an approximate result. The advantages of this method are: a) It gives an approximate result by simulating the subconscious human way of thinking. b) It successfully combines the vague information taken from a picture with the error tolerance result. c) It gives a quick response, since, in discrete analysis, the results of all input combinations can be pre-calculated and stored as a matrix. 2.IMAGE PROCESSING The purpose of the image processing unit is to identify the fruits in the picture taken from the camera. Specifically the position, the size, and the boundary of each fruit that appears in the picture is determined. It was found that image analysis of 64x64 pixels with four bits (16 grey levels) was adequate for our experiment. A powerful source of light is used from the point of observance in order to eliminate shadows on the fruits that may be created by natural light, and to make the system able to function during the night. The location of the fruits is determined in two stages using a modification of the region growing method [3,7]. Initially, certain "seed areas" are chosen (using appropriate criteria) which in turn are enlarged step-by-step. At each step, if a pixel neighbours an area and fulfils certain requirements, it is embodied and the procedure is then repeated. In this case the image processing unit locates the areas of high density. This is because, due to the way of lighting, the intensity is high at the central area of the fruits, which gradually decreases towards the boundary surface. After this, at every stage of enlargement, the neighbouring pixels of each area are checked. If the intensity of a certain pixel is less or equal to the intensity of the neighbouring border pixel, the pixel is embodied in the main area. This process is repeated until the enlargement of the seed area is completed. It is noted that with this procedure, all areas of high intensity are located (not only the fruits), i.e. parts of the sky (during the day) as well as the reflection of light on leaves. Based on the smooth changes of intensity of the areas which represent the fmits in contrast with a sudden change of other areas, the program distinguishes which of the located areas are actually fruits and which are not. 3.BRIEF REVIEW OF FUZZY LOGIC 3.1 Fuzzy sets A fuzzy subset A of a Universe of Discourse U is characterised by a m^embership function I: U -> [0,1] which associates with each element x of U a number l\{x) in the closed interval [0,1] which represents the grade of membership of x in A. The fuzzy subset A will be denoted as:

555 n

i=l

where E stands for union. For example if U = 0+1+2+..+9 then a fuzzy subset could be ; A = small digits = 1|0 + 1|1 + 1|2 + 0.7|3 + 0.3|4 + 0|5 + ... + 0|9 where the symbol (+) stands for union. Three basic operations on fuzzy sets are: i) The union of fuzzy subset A and B of Universes of Discourse U is denoted by AUB with a membership function defied by : ^'AUB(X) = inax[

H\(x),

HB(X) ]

, x€U

This corresponds to the connective "OR" ii) The intersection of the fuzzy subsets A and B is denoted by A^B with a membership function defined by »*AnB(x)

=

"Jn[

MA(X).

^B^*) 1

, xCU

This corresponds to the connective "AND". iii) The complement of the fuzzy subset A is denoted by -,A with a membership function defined by: H-.A^'^) = 1 - HA(X)

, x€U

3.2. Fuzzy conditional statement and compositional rule of inference. Let A and B be fuzzy subsets of two different Universes of Discourse U and V, respectively. Consider the fuzzy conditional statement (Linguistic implication): R: "if A then B" or "A => B" The relation R is a fuzzy subset of the cross product U X V with a membership function defined by [6] MR(x,y) = min[ fi\{x),

fiB(y) 1

, xCU,

y€V

Given a fuzzy relation R and a new fuzzy subset A' (the antencent) the consequent B is inferred from the compositional rule of inference [5]: B' = A' 0 R

556

The membership function is defined by : ftB'(y)

= maxinin[

\i\{x),

HR(x,y) ]

, xeU,

y€V

X

Finally two or more rules R; can be combined using the connective ELSE to give a fuzzy algorithm R which describes the relation between linguistic variables leading to an approximate result. The algorithm R is given by the relation: R = R l u R2 U R 3 u . . u R j 3.3.

Defuzzification

The result which is given by the a fuzzy algorithm is a fuzzy subset B'. Various procedures may be used to select a certain value for the output. Here, the element with the maximum membership value is chosen. In case there are more than one elements with maximum membership value in B' the average of the maximum and minimum of these elements is chosen.

4. THE PROBLEM OF THE DIRECTION OF APPROACH Initially, Cartesian co-ordinates are defined as shown in fig.l with centre of axis where the bunch and the branch connect. This point, due to the similarity of the fruits, is considered a little higher than the bunch 's centre of mass

Fig.l. Direction approach: Cartesian coordinates In order to infer the direction of approach two angles need to be determined. Using spherical coordinates (r,cp,8) these angles are cp and 6, where 0 is the angle between the direction of approach and z-axis. The gripper must approach the fruit towards the direction of the vector which begins at the fruit's centre and ends at its point of connection with the branch. The projection trace of this vector on the x-y plane is approximately

557

considered as the section connecting the centre of the fruit's image (Kx,Ky) with the considered point of connection. Thus, the angle q) is determined by : Ky tf= arctan Kx If the centre of the fruit's image is to the right of the point of connection, then the solution in the interval (-n/2, n/2) is accepted, otherwise the solution is sought in the interval (jt/2, 3jt/2). The angle 8 is determined through the fuzzy algorithm. The input variables here are the following: a) The visible percentage of the fruit from the observing point. b) The distance of the images centre from the point of connection with the branch. For example if the fruit is fully visible and the point of connection lies behind it then the approach should be attempted from the front side (6=0 ). On the other hand if the point of connection lies towards the fruits outermost points then the approach should be attempted from the side (6=90 ). The entire fuzzy algorithm consists of seven rules. The Universe of Discourse of the variable "Area" (A) is: U = 1 + 2 + ... + 10 where the value 10 specifies a fully visible fruit. Three labels were chosen: SmalUS), Medium (M) and Large(L). The appropriate sub-sets are given in Table 1. Table}: Support subsets for the labels of variable "Area". Area small med. large

1 10 0 0

2 7 0 0

3 3 3 0

4 0 7 0

5 0. 10 0

6 0 7 0

7 0 3 3

8 0 0 7

9 0 0 9

10 0 0 10

The universe of Discourse of the variable Distance (D) is V = 0 + 1 + 2 + ... + 10 The "Distance" may be: Short(S), Medium(M), Long(L). The support subsets are shown in table 2. Table 2: Support subsets for the labels of "Distance" D short med. long

0 10 0 0 0

1 9 0 0

2 7 3 0

3 3 7 0

4 0 0

5 0 10 0

6 0 7 0

7 0 3 3

8 0 0 7

9 0 0 9

10 0 0 10

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The variable "Angle" (8) is quantised into 13 levels ranging from 0 up to 180° (every 15°). The Universe of Discourse is W = 0 + 1 + 2 + ... + 12 Seven normalised labels where chosen: ©1 (front) 92, 0 3 , 0 4 (sidewise), 05, 06, 0 7 (Back). The corresponding support subsets are given in table 3. Table 3: Support sets for the labels of "Angle"

e

»i 02 ®3 ©4

es ©6 07

0 10 3 0 0 0 0 0

1 7 7 0 0 0 0 0

2 3 10 3 0 0 0 0

3 0 7 7 0 0 0 0

4 0 3 10 3 0 0 0

5 0 0 7 7 0 0 0

6 0 0 3 10 3 0 0

7 0 0 0 7 7 0 0

8 0 0 0 3 10 3 0

9 0 0 0 0 7 7 0

10 0 0 0 0 3 10 3

11 0 0 0 0 0 7 7

12 0 0 0 0 0 3 10

The fuzzy algorithm consists of the next seven rules: IF A=Large AND D=Short THEN e=Very 0 1 ELSE IF A=Large AND D=Medium THEN 6= 0 2 ELSE IF A=Large AND D=Long THEN 0= 0 4 ELSE IF A=Medium AND D=(Short OR Medium Long) THEN e= 0 3 ELSE IF A=Sman AND D=Short THEN 9= 0 7 ELSE IF A=Small AND D=Medium THEN 8= 0 5 ELSE IF A^Small AND D=Long THEN 8= 0 6

OR

For the single fruits the following values are chosen: q)=270O

, 8=450

taking into account the possibility that the second fruit may be hidden behind the visible fruit.

559

5.RESULTS The values table: ^'A 0 1 2 3 4 5 6 7 8 9 10

of 9 for all combinaiions of the input variables are given in the next look-up

173 173 165 143 135 135 135 128 120 120 120

1 165 165 165 143 135 135 135 128 120 120 120

2 3 4 135 120 135 120 135 120 135 120 128 120 128 120 128 120 128 120 120 120 120 120 120 120

5 120 120 120 120 120 120 120 120 120 120 120

6 120 120 120 120 120 120 120 120 120 120 120

7 75 75 75 75 83 83 83 83 105 105 105

8 8 8 8 38 45 45 45 68 90 90 90

9

1 8 8 8 38 45 45 45 68 90 90 90

0 8 8 8 38 45 45 45 68 90 90 90

Two examples of the program's execution are given bellow. The first picture (a) is the image taken from the camera, the second (b) depicts the values of intensity of every pixel, the third (c) gives the "seed areas" and the fourth (d) providesthe areas which have been located as fruits. Values of intensity greater than nine have been replaced with the value of nine for reasons that are connected to the representation of the picture. For the same reason, in pictures (c) and (d) intensities of zero value are not typed. 1) First example:

(a)

(b)

(c)

(d) Fig.2: First Example

560

The program locates a single fruit on the left side and a branch consisting of three fruits on the right.Results; Single fruits: f=27l* , 8=45® Bunch; Fruit No 1: f=-f« , §=90® Fruit Mo 2; f=189« , i=l§5<» Fruit No 3: f^-SS® , i=12iO 2) Second example:

(b)

la}

.

" j i - J , - ----J - • ] .

ic)

(di

Mg.?- Secotid E.«iump!e The program locaies live iruiis in two bunches; Tlie upper bunch c^.n.si^ts of ibrcc frjits and the Results: Upper hunchfirst fruit. second fruit : third fruit: Lower bunch:

first fruit: second fruit:

lower one consists of two fiiiits: ip^fo , f=172'' , f=2630 ,

i=f§o e=fi* i=12§o

f=2i7» ,

f=l©50

561

It is noted, that after the removal of a fruit from a bunch, the positions of the other fruits change and the procedure must be repeated. 6.CONCLUSIONS This paper, belongs to the area of designing robots that function in the natural environment. Such robots are equipped with vision and artificial intelligence. The problems that are faced in the attempt of locating the fruits are many, since in each case the position of the fruits in the picture are random and the background different. Through the study of ways to surpass these problems, useful conclusions of the importance of characteristics, such as colour, shape and intensity, in order to locate the fruits, are drawn. The principal issue of the present paper is the fuzzy algorithm. The central problem in constructing this algorithm was the description in linguistic terms of the way the human infers in which direction the fruit must be detached from the bunch. Since this problem was surpassed the fuzzy algorithm was easy to construct. This is actually the main advantage of fuzzy logic, i.e. from the moment the control strategy can be expressed linguistically as a set of fuzzy conditional statements, the fuzzy logic system is readily synthesized. Here, the fuzzy algorithm was shown to be of a greater success than expected, encouraging future studies of the fuzzy logic in image processing too. In conclusion fuzzy logic theory proves to be a helpful tool in the human attempt to create robots with "human" abilities such as fruit's collection, etc.. REFERENCES [1] S.G.Tzafestas .: Expert Systems in Engineering Applications, Springer Verlag, Berlin, 1993. [2] S.G. Tzafestas.: Intelligent Robotic Systems, Marcel Dekker, New York, 1991. [3] R.G Gonzalez and R.E. Woods. : Digital Image Processing, Addison Wesley, pp.574-5,I992. [4] LA. Zadeh: Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans. Syst., Man, Cybern.,vol.SMC-3,no.l,pp2844,Jan.l973. [5] K.Hlrota, Y.Arai and S.Hachisu: Moving mark recognition and moving object manipulation in fuzzy controlled robot. Control: Theory and Adv. Techn.,vol 2, no.3, pp 399-418, Nov.1986. [6] E.H.Mamdani: Application of fuzzy logic to approximate reasoning using linguistic synthesis. IEEE Trans. Computers, vol c-26,no. 12,Dec.l977. [7] K.S.Fu, R.C.Gonzalez, C.S.G.Lee: Robotics: Control, Sensing, Vision and intelligence. McGraw-Hill 1987. [8] R.M.Tong: A control engineering review of fuzzy systems. Automatica,vol.l3,pp.559-569.1977.