On stability of formal fuzziness systems

INFORMATION SCIENCES 22, 51-68 (1980)

51

On Stability of Formal l~"~ness Systems ALEKSANDER A. KANIA

Institute of Applied Mechanica, Mathematica Dioiaion, Technical Unit~raiOpof laelce, AL Ty~'~lecia Pa~twa Polakiego 7, 25-314 Kielce~ Poland and JERZY B. KISZKA, MARIAN B. G and MARIAN S. STACHOWICZ

O

~

,

JAROSLAW It. MAJ,

Institute of Automatic Contro~ Technical Unioer$iO~of lOeic~ AL Tys~lecia P~mstwa Polakiego 7, 25-314 Kielc~ Poland Communicated by L. A. 7adeh

ABSTRACT The structure of formal fslT'~inesssystems as some abstraction on real f-~,iness systems is debated. Then the global definition of a-stability is given. First some necessary and sufficient conditions for a-stability are given for the case of R .. A ~ B (Sec. 2) and later for the case of R defined by an implication chain of any finite length (Sec. 4). In Secs, 5 and 6 notions of a-stability and a-fl strong decision stability are introduced, and some theorems on necessary or sufficient (or both) conditions for these kinds of stability, imposed on the structure of an impfication chain, are proved. In Sec. 3 the good-mapping property of a fuzzy relation matrix is a n M y ' / ~ and because of its heuristic importance it is assumed in further ,.,ections. In many cases examples are given to illustrate definitions and theorems or conclusions.

1. FORMAL FUZZINESS SYSTEM We ~ssume that the definition of a fuzzy relation is known to the reader. Thus we shall devote our attention to the method of construction of a fuzzy relation matrix. As pointed out in [6], this can be done by an implication or by a chain of implications. N o w we shall briefly recall what is known. Let X and Y be some finite sets, cardXffi n, card Yffi m. Let A and B be fuzzy sets defined on X and I,', respectively. Then A ~ B ~ A x B (where x is the Cartesian product in the sense of fm,~iness) can be treated as the definition of R: a fuzzy relation. If one takes under consideration a family of fuzzy sets ©Elsevier North Holland, Inc., 1980 52 Vanderbilt Ave., New York, NY 1 0 0 1 7

0020-0255/80/07051-18501.75

52

A L E K S A N D E R K A N I A ET AL.

At . . . . . A, defined on X a n d a family of fuzzy sets Bi . . . . . B, defined on Y, one can construct a chain of implications (AliBi) or ... or (A,~B,). By definition, the length of this implication chain is s > 1. Then in the preceding case, R=(A~B), R was defined by an implication chain of length 1. In the latter, the chain of length s can also form a relation R. In what follows, when we say "relation," we mean a chain of implications of length s, s • 1. A fuzzy relation R is a form of mapping with its d o m a i n in ~ , a family of all fuzzy sets defined on X, and a space of values contained in ~ , a family of all fuzzy sets defined on Y. This mapping can be denoted in terms of the compositional rule of inference (see Z a d e h [1]):

UoR=V, where U is any fuzzy set defined on X, and V any fuzzy set defined on Y. This mapping can be treated as a model of a real transformation system which consists of a fuzzy input, black box and fuzzy output:

Fuzzy input--. ~

--,fuzzy output.

W e assume that this real transformation system can be m a p p e d on to the formal model:

u--,®--)v

(,)

or equivalently,

UoR=V, where U and V are also fuzzy input and output, respectively, but they are formally well defined. Similarly R is not a mere idea of fuzzy transformation, but a real matrix, suitably defined. U a n d V are abstractions on the real fuzzy input a n d output, a n d R is an abstraction on a black box, a natural transforming system. Then the formula U o R = V, until you hear further, will be called a "formal fuzziness system." In the present p a p e r we are not interested in studying the adequacy of relation R with respect to a real transformation system. But when such adequacy occurs, some problems of great importance in real transformation systems are converted to corresponding problems in their formal models. One of them, the problem of stability, is a subject of study in the present paper.

ON STABILITY OF FORMAL FUZZINESS SYSTEMS

53

As far as we know, stability has been considered only in [2,3,4,5]. As Negoita [4] maintains, a system has the stability property if the state trajectory can be held inside some given domain for some nonempty set of inputs. This is consistent with the classical definition of stability, but his insight into the problem merits for further attention, especially in view of its possible applications. Mamdanl [3] strongly stresses the importance of stability problem for fuzzy system both in practice and theoretical debate. But in his opinion, this problem has not been satisfactorily stated, and therefore has not been solved yet. The main reason, in his view, lies in the absence of a developed theory for the design of fuzzy systems--in particular, in the absence of good theoretical tools for studying fuzzy-system stability. Tong [5] defines a fuzzy system by means of fuzzy sets of finite dimension. Stability of fuzzy system is defined in terms of an equivalence relation in the space of fuzzy sets. This relation is generated by the peak-pattern function. This attempt is applied to closed-loop fuzzy systems and arises in connection with the controLlability of a system. It is of great practical importance, but the application of the peak-pattern criterion restricts studies on other aspects of stability of fuzzy systems. Kickert and Mandani [2], recalling the classical approach, treat the fuzzy system as multilevel relay. Applying the described function method, the relay's stability can be investigated; thus investigation of the stability of a fuzzy system, under some assumptions, is made possible. These assumptions, however, seem to be very restrictive for practical purposes. This short review suggests that research on stability is caLled for, but up to now there have been but a few commonly accepted definitions and criteria for those fuzzy-system properties which can be compared with stability in the nonfuzzy case. The present paper is an attempt to jump over the gap dividing fuzzy and nonfuzzy analysis of transformation systems. 2.

a-STABILITY, T H E CASE OF SIMPLE IMPLICATION

At the be#nning let us consider a formal fuTziness system in a special form. That is, let U, V be nonfuzzy sets, and R a nonfuzzy mapping R : ~ ( X ) - - , (Y). Let Y D F(Y) be the set of feasible points in Y. Then we consider the system (*) as having the stability property with respect to some subfamily P ( X ) C ~ (X) if for each U ~ P ( X ) , U o R N [ Y - F( Y)] = 0 . In other words, the system has the stability property with respect to some class P ( X ) if for each input X E P(X), X o R does not contain any non.feasible point. This definition of stability in nonfuzzy systems can be generalized to include also the fu77iness case,

DEFINITION 1. The formal fuzziness system has the a-stability (a-S) property with respect to some family t~ c ~(~ and some feasible subset F(Y) C Y if

54

A L E K S A N D E R K A N I A ET AL.

~.0.

or4

I% '

~tY)

4,

~r Y- F(Y~

Fig. I for every C D U fuzzy set, U o R = V is such that t t v ( y ) < a for every y E Y F(Y). E X A M P L E 1. F o r some input C ~ U, let the output set V have the m e m b e r ship function shown in Fig. 1. W e see that in this case the system has not the a-S property with respect to the family ~ a n d given F(Y), because for some U E C, U o R = V is such that there is a point, say Y0 E Y - F ( Y ) , such that #v(Y0)>a. Of course, in this example a~-stability (a < a t ) is not excluded; however, it can h a p p e n that there is another set W E A such that m a x y _ ~ r ) l z w . R ( y ) > a I. This would imply that the system ( * ) is not a i-S with respect to the given family A a n d feasible subset F ( Y ) . In the case of the nonfuzzy system ( * ) it is obvious that its stability implies a-stability for any a E [0, 1]. 2.1. SUFFICIENT AND NECESSARY CONDITIONS FOR a-STABILITY OF A SYSTEM (*) Let the fuzzy relation matrix R, R = A ~ B = A × B, be given. Elements of this matrix can be written as follows: r#=l~A(xi)AuB(yj), i = 1 . . . . . n a n d j = 1 . . . . . m. Then any output V, associated with an input t) can be characterized as follows: #v(Yj) -- VT-I tZu(Xi)Aro" -~ V~ l I ~u(Xi) AU~(Xl)Au~(Yj) ~ ~B(Yj) A k/7.11~v(xi)Aua(xj), j = l . . . . . m. It can readily be seen that uv(Yj)
#B(Yj) A V # u ( x i ) A u A ( x i ) < a . i--I

(1)

ON STABILITY O F F O R M A L F U Z Z I N E S S SYSTEMS

55

If I~n(yj)
Proof. Sufficiency is an immediate consequence of the inequality (1) and the observation below it. For the necessity, let VT_l/La(Xs)>a. We will prove that gB(Yy)< ~ for all yj E Y - F ( Y ) . Given a set A, we can always choose a set U~% such that gv(x~)>g,~(x~). This means that V T . i g v ( x , ) A g a ( x i ) > VT-~l~A(X~) >a. On the contrary, suppose for some)9 ~ Y - F ( Y ) that gn(Yy)> a is true. Then because of (1), V'2. tltv(Xi)AgA(Xl) VT.~gv(x~)Aga(x~) = VT- ~g,~(xi). • TrmoPa~M 2. Consider a system where R = A ~ B, and some family ~ of fuzzy sets on X. Then V T . tgtl(xi)AgA(Xi) a , then % D A = { U: ff gx (xi) '; a then gu(xi) E [0, 1] else ~tu(xi) < a } .

E X A M P L E 2. Let X = (1,2,3,4}, Y-- (1,2,3,4,5}, and

A = T1 + _0~ + _0~ +~0 411

0.8

0.3

0],

0

0.2

0.3

0.7

1

~[0

0.2

0.3

0.7

1],

B= T +-~--+-~ + - T + $

56

A L E K S A N D E R K A N I A ET AL.

A=~B= R=

0 0 0 0

0.2 0.2 0.2 0

0.3 0.3 0.3 0

0.7 0.7 0.3 0

1 ] 0.8 0.3 " 0

]

Let F ( Y ) = {1,2,3}, then Y-F(Y)={4,5}. N o w we shalldefine a family A under which the system (*) specifiedabove has the 0.3-S property. Because the condition in (a) is not satisfied,from Co) we get 6)(, D t~ = ( U : # u ( x t )

< 0.3p, u(X2) < 0.3/.tu(x3) E [0, 1], #u(x4) E [0, 11 ).

For example U--[0.1 0.2 0.8 0.1]EA and in fact UoR=[O 0.2 0.3 0.3 0.3] satisfies the a-stability condition. If, with respect to some particular demands, the operation A is defined as an algebraic product, then if point (a) cannot be considered, we define a family with respect to the description given in Co) as follows:

% ~ ~ = { U: ~u(X,) < ~

if ~,(x,) ~ o

or #u(xi)= I if/~A(xi)=0). Of course, ~

C ~.

a) to

,I.o

08 O.6

Q~

0.~

o2

02

4

Fig. 2

2

3

~

5

ON STABILITY O F F O R M A L F U Z Z I N E S S SYSTEMS 3.

57

GOOD-MAPPING PROPERTY

In this and the following sections we shall concentrate on systems (*) with R defined by an implication chain of length greater than 1. At the b e # n n i n g we consider the case of length 2, i.e. R = ( i f A then B else C ) = ( A ~ B + n o n A =~C). The matrix R has elements r¢ which can be written as r# ~- pA(xi)Ap.B(yj) V(1 --I~A(x~))Ap.c(Yj), i = 1. . . . . n , j = 1. . . . . m. If we choose as components of the relation matrix arbitrary fuzzy sets A, B, C, then we do not always have A o R = B or nonA o R = C. Let us consider the following example: E X A M P L E 3. Let X, Y as in Example 2, A =[1 0.6 0.3 0], B----[0 0.2 0.4 0.7 l], C = O 0.8 0.6 0.3 0l,

R_-

0 0.4 0.7 1

0.2 0.4 0.7 0.8

0.4 0.4 0.6 0.6

0.7 0.6 0.3 0.3

1 ] 0.6 0.3" 0

One can compute that AoR=[0.4

B=[0

0.4

0.4

0.7

1],

o.z

o.4

0.7

1].

but

Then A o R ~ B for the boldface components. Similarly nonAoR---[1

c=[l

0.8

0.6

0A

0.4],

o.8

0.6

0.3

0 ].

but

The same can be seen in Fig. 3. In Example 3 we encounter a difficulty. The matrix R defined on the sets A, B, C does not repeat the transformation. Verbally R can be described: if A then B, and if nonA then C. But for the superpositions we have A o R ~ B and nonA o R ~ C . This lack of consequence conflicts with the need for consistency with reality, which requires some kind of "repeatedness." This we shall introduce as the "good-mapping property" of the relation R. DEFINmO~ 2. Let R f ( A l ~ B O o r . . . o r ( A , ~ B ~ ) be given. We say that R has the good-mapping (GM) property iff for every i-- 1..... s, Al oR ffi B s.

ALEKSANDER KANIA ET AL.

58

jlll~ tO

~,0

08.

O8

0.6

O~ a~ 02

0.2

y~

Fig. 3

Of course the GM property of a given matrix R depends on the choice of fuzzy sets AI, B~. Now we shall be studying the properties of sets A~,B~ which are of importance to the presence or absence of the GM property. PgOPOSlTIOI~ 1. Let R = A ~ B e l s e C be give~. Then #A.R(yj)=l~B(yj) if /~c(YJ) < /~(YJ) and/~B(yy) < max,/tA(x;).

Proof. Surely/~,~(x~)> [1-/~a(x~)]A/~A(xi). Combining this with the supposition Pc(Yj)
/,LA(xi)A/~B(Yj) ;~#a(xi)A[ ! --/~A(Xi)]A/~:(yj). Let us notice that ~Ao,~(Yj) =

~/ {/.LA(X~)A/-LB(yj)V/~A(xt)A[1--/~A(Xi)]Ap, c(Yj)}, i--I

j ~ l~...~m. Taking into account the previous inequality, we get pA.R(yj)ffiVT. IpA(xl)A /LB(yj), but this means that I~AoR(yj)-~I~B(yj)AVn. II~A(Xi)mI~B(Yj). The last equality is the consequence of the third supposition /LB(yj)/tB(y3); this means that the converse implication to that formulated in Proposition 1 is not true,

ON STABILITY OF FORMAL FUZZINESS SYSTEMS

59

Proposition 1 can be symmetrically formulated for the case of fuzzy sets nonA and C: PRol'osmoN 2. Let R f A f , B else C. Then /~o,A.~Os)fp~c(yj) if Pc(Yj) "

max~~..~(x,) and ~,(yj) < ~c(Yj). The proof is the same as for Proposition 1; it suffices to substitute nonA for A and C for B. Example 3 illustrates this case, too. Note that lta(yO ~ Pc(YO and FB(Y2)~ Ps(Ya)fPc(Y3), which implies that p~o=A.R(yj)= Pc(YJ) for j,= 1,2,3. Both propositions can be generalized for the case of a fuzzy-relation matrix defined by an implication chain of any finite length. TEmoI~M 3. Let A i ..... A, be fuzzy sets defined on X, Y.~. l pa,(x) = I for any x E X , and let Bj ..... B, be fuzzy sets defined on Y. Let Rffi(A~=~Bt)or...or ( A , ~ B , ) . Then /~A,.R(yj)f/%(yj) if for some k E { 1 ..... s} and some j E { 1..... m} we have tta,(yj)< max x pa~(x) and FB~(Yj)> lta,(Yj), l ~ k . The proof is analogous to that given for Proposition 1: Surely /x~(x) for l¢=k. The second supposition gives gA,(x)Apa,(yj)~ttA,(x)A t%(x) A tts,(Yj). By definition tt~,.R(y/) ffi VT- ~ ( ttA,(x3 m g.~,(x3 A ttB,(yj)) V . . - V(pA,(x,)A/~B,(yj)V... V[ l.tA,,(xi)ApA,(x,)Altm,(yy) ]. Now, on the basis of the inequality stated it is obvious that

ttA,(x)At%(x)

i--I

= ~B,(yj)A ~/~,(x,) = ~,,(y~) i--I

on the strength of the first supposition.

•

Thus we have related a sufficient condition for the equality p~.s(yj)== Y. If this inequality holds for any yj E If, the matrix R has the GM property, as stated in definition 2. Now we shall give a sufficient condition covering this case. Let us return now to Example 3. We redefine sets B and C as follows: B==[0.4 0.4 0.4 0.7 1], C=[1 0.8 0.6 0.4 0.4], while A remains unchanged: A==[1 0.6 0.3 0]. Then

IZBt(yj) for some yj E

(A~BelseC) f R=

0 0.4 0.7 1

0.4 0.4 0.7 0.8

0.4 0.4 0.6 0.6

0.7 0.6 0.4 0.4

1 ] 0.6 0.4" 0.4

60

A L E K S A N D E R K A N I A ET AL.

One can now check the following equalities: A . R - - - B and nonA .R--- C, that is, ttA.s(yj) ffi Pn(Yj) and tt.,,.A.s(yj)ffi tts(yj) for a n y y j E Y. This means that for the fuzzy sets A,B, C so defined, the matrix R has the G M property. The way of redefining component sets is elucidated in the following proposition: PROPOSITION 3. Let R ----(A ~ B else C). I f PB(Y) ~ maxx PA(x) and PB(Yfl < Pc(Yj) implies I~,(Yj) • maXx/~a(x)A[l -- la,~(x)], then pA. a ffi ILn.

Proof. Let us take any yj E Y. If l~n(Yj) • ttc(Yj), then as proved in Proposition 1, p,~.a(yj)ffip~(yj). If on the contrary pB(Yj)
~,, (x,)A#, (yj) • ~A(x,)A[l - ~A(x,)], and finally

pA(xs)ApB(yj))/tA(x;)A[! -- pA(x,)Attc(Yj)].

This implies that

n

#~.~(y:) ffi V #,,(x,)A~B(yj) i--I

ffi ~,,(yj)A ~ / # ~ ( x , ) ffi # , ( y j ) i--I

as a result of the first supposition.

•

As a generalization of Proposition 3 for the case of an implication chain of finite length we can state the following theorem:

TrmORr_~ 4. Let Rffi(Al=*BOor...or(A,=~B,). Then/~,4,.R ffi/x~ if (1)/~a,(Y) •.maXxpa,(x ), and (2) #B,(yj)
max~ { # ~ ( x ) ^ I 1 - ~(x)]}. In terms of this definition w e can generalize Proposition 3 as follows.

THEORmCI 5. Let A t ..... A, be fuzzy sets defined on X, E'l_llXA,(x) ffi I for any x E X , and let Bt ..... B, be fuzzy sets defined on Y. Let R - - ( A l = ~ B O o r . . . o r

ON S T A B I L I T Y OF F O R M A L F U Z Z I N E S S SYSTEMS

61

(A,~B,). Then #A,.~ =#n, if (I) ttn,(y ) < maxxtxA,(x ) for i-~ 1..... s and any y ~ Y, and (2) ItB,(y) <#~(y) for some y E Y and some j ~ i implies p(Ai) < ~Bi).

Proof. We shall prove that the inequality p(A~) ,; p(B~) implies the inequality #a(Y) ~ maxx #A,(X)A#,~j(x). It is obvious that lxa,(y ) ~ K Bl) ~ p(A,); then Ixn,(y) ~=o(Ai). In consequence 1 -/~B,(Y) '; 1 - p(Ai) - 1 - max x { #a,(x)A [1-/XA,(X)] } = m i n x ( [ 1 - gA,(x)]V/ttA,(X)}. OIl the strength of the orthogonality assumption, we have #~,(x) < 1 -- #a~(x), provided that we have the inequality

x

= ~ - m~,, [ ~,(~)A~M) x

].

Therefore 1 - ixn,(y)= I - m a x x [ ttA,(x)A/LAj(X)], which is equivalent to

but this, according to Theorem 5, suffices to finish the proof. The matrix that of and Ai 4.

•

theorem can be summarized briefly in this fashion: a fuzzy-relation R has the G M property if the grade of fu7yiness of Bl is not less than As except for the case/~a, > ~ f o r j ~ i . In this ease no restriction on Bj to maintain the G M property of R is necessary.

a - S T A B I L I T Y I N T H E CASE O F A C O M P O S I T E R

By a composite R we mean a relation R defined by an implication chain of length greater than 1. Let us start discussing a-stability with the following theorem: THEOREM 6. Let fuzzy sets A I..... A, and BI ..... B, be defined on X and Y respectively, and R = ( A t ~ B O o r . . . o r ( A , ~ B , ) . Then for any fuzzy U defined on X, the following inequaliO, holds:

62

ALEKSANDER K A N I A ET AL.

Proof. By definition Pu.R(Y) = V~- ~[ ~u(xt)ApA,(xa)ApB,(y)]V" • • V[ Pu(Xj) Aps,(y)ApA,(x)]. Let us note that for any iffi I ..... n, [pu(xi)ApA,(xi)ApB,(y)] V . . . V[ pu(xt)Aps,(x~)ApB,(y)] ~ V~ - ~P~,(y). Then this inequality holds also when maximum operator V is used, which completes the proof. • In terms of Definition 1, the preceding theorem gives the following conclusion with regard to ,,-stability.

THEOREM 7. Let Y D F(Y) be the feasible set. Then system (*) has the a-S property with respect to 96, a fami~ of all fuzzy sets defined on X, if for any

y~. Y - F(Y),

V~_lp~.(y)


Theorem 7 deals with any system U o R - - V , where R is defined by any implication chain of finite length. In what follows we turn our attention to the case with R having the GM property, as defined in Sec. 3. Especially we will be concerned with some class of fuzzy relations R as defined by a chain (A~=* BOor...or(A:*B,), with the family At a noufuzzy disjoint cover of X, and Bt fuzzy sets defined on Y. It can be easily proved by use of Theorem 5 that R so defined has the GM property. It suffices to note that p(At)-" 0 ~ p(Bt) for Bt any fuzzy sets contained in Y, and At nonfuzzy sets contained in X. It may seem that restriction of attention t o nonfuzzy entries At leads to triviality. But, as a matter of fact, we put no restriction of the input U of the formal fuzziness system U o R ffi V. U can be either nonfuzzy or strictly fuzzy. Thus our restriction is of importance only at the stage of synthesis of the fuzzy relation matrix R, and not at the stage of exploitation of the system (*). One can presume that precise identification of the fuzzy relation may lead to description of inputs Ai as non.frizzy, while outputs Bt can be defined arbit r a r i l y - n o t violating, however, the GM property. In the case under consideration Theorem 6 is of course valid. Some special results of this theorem can be seen in the theses of the following proposition. PROPOSITION4. Let R ----(A i ~ Bi) or... or (As~Bs) , U a non fuzzy subset of X. Then pv.s(y)ffiViEtps,(y) if U N A i ~ O for i E l . In particular, if UffiffiX, then Pu.R = V[-IPn,.

Proof. pu.R(y ) ffi VxeA, [ Pv(x) A psi(y)] V" • • vVxeA" [ pu(X) A Ps,(Y)] PB,(y)AV~A, Pu(x) v ' ' " V P n . ( y ) A V ~ A . pu(x) = V , ~ , p s , ( y ) A V . ~ . 4 , pu(x) V t e t Ps,(Y), because 1 ffi V~ cA, pu(x) for i E I, while V~ ~Ajpu(x) •D for j ~ which completes the proof.

ffi ffi

I, •

The following example will cover the case of this proposition. EXAMPLE 4. Let X, Y,B, C be as in Example 3. A =[1 0 0 0], nonA ----[0 1 1 1], U = [ I 1 0 0].Of course U f - l A f f i ( y l ) ~ O a n d Uf'lnonAffi{y2}~O. Then U.Rffi[I 0.8 0.6 0.7 I]ffiBUC, as is consistent with the proposition proved.

ON STABILITY OF FORMAL FUZZINESS SYSTEMS

63

Theorem 7 covers a special case when the system (*) has the a-S property with respect to the full family %. Now we shall prove necessary and sufficient conditions for a system to have the a-S property with respect to some family d~ of fuzzy sets defined on X. In the following theorem it is assumed that R ffi( A l ~ B 0 o r . . . o r ( A , ~ B , ) , and the Al, as stated earlier, are nonfuzzy. THEOREM 8. Let A be some family of fuzzy sets defined on X, and F( Y) a feasible subset of Y. Then the system ( * ) has the ot-S property with respect to the family ~ if and only if the inequality/xB,(y ) > ct for some y ~ Y - F ( Y ) and some i E { I ..... s} implies the inequality Vxea, lXu(x) < ct for any U E ~.

V.

Proof. Let y ~ Y - F ( Y ) ; " v v x ea.l~v(X)Al~B.(y ).

then

u v . R ( y ) = V x e a m # u ( x ) A/~B,(y )

Necessity by contraposition: Let V.eA, I~u(x) > a for some A i. Then, assuming the tx-S property,

a )#UoR(y ) > V

P'u(x)AI~B,(Y)

xEAI

=/'LBI(Y)A V p.u(x), xEA

I

which implies that/~B(Y) < a for a n y y E Y - F(Y). Sufficiency: By assumption, for any i = 1 ..... s and any y E Y - F ( Y ) we have/LB,(y)AVxea, IXv(x) < ct. Thus the maximum operator does not violate the required property. So

xEA

I

xEA,

for any y ~ Y - F ( Y ) . But this coincides with the definition of a-stability, which completes the proof. • Let some fuzzy-relation matrix be given. Then Theorem 7 can be used for an operational characterization of the family 6g such that R has the a-S property just with.respect to A. Namely:

l~={UfuzzyonX:forsomeyEY--F(Y),l~B,(y)>a~

V /~(x)
I

The following example gives an illustration of how U should be examined to decide whether it belongs to d~ or not.

64

ALEKSANDER K A N I A ET AL.

EXAMPLE

5. Let X--{I,2,3,4}, Yffi{I,2,3,4,5},F(Y)--{I,2,3}, and R--

(A~BelseC) for some nonfuzzy A and some fuzzy B,C. The family ~ can be described as follows: = { U fuzzy on X:#n(y~) > a implies V tLu(x)
and/tc(yj)>aimpliesa>

~ x E

#u(x)forsomei,jE{4,5}}.

nonA

E.g. let A ffi[1 0 0 0], Bffi[O 0.2 0.4 0.7 1], Cffi[l 0.8 0.6 0.3 0], affi0.3; in this case pn(y4)ffi0.7>0.3 and pB(ys) = 1 >0.3. This implies, according to the given rule, that ~/x~A #u(x)ffi g u ( x l ) < a =0.3. On the other hand,/tc(y4) 0.31, a and/~c(Ys) ffi0 < a, and this implies that no restrictions must be put on values o f / ~ ( x i ) for iffi2,3,4. Thus, given a fuzzy relation matrix R, the family under which R has the a-S property consists of all fuzzy sets U such that p ~ x i ) < 0 . 3 , gu(x~)E[0,1] for iffi2,3,4. For instance Uffi[0 0.9 1 0.2] and W - [ 0 . 2 0.4 0.8 1] belong to the family ~ because they satisfy the conditions formulated above. In fact one can check by computation that U o R = [ I 0.8 0.6 0.3 0] and WoR=[1 0.8 0.6 0.3 0.2]. In beth cases the last two components, corresponding to a nonfeasible set {Y4,Ys}, are not greater than a ffi 0.3. =

5.

a-DECISION STABILITY

Continuing Example 5 of the last section, let us consider the output of a system if the input U = [0 0.2 0.3 0.2]. Obviously U o R--[0.3 0.3 0.3 0]. This means that U E ~ , but at the same time 0.3 is the maximum value of the membership function. It happens in decision procedures that we must choose one point from X having the highest membership in an output fuzzy set. In our case this maximum is 0.3, and it is also the barrier point because of stability. This maximum can be a negligible quantity. This can happen if one wants a strong maximum to lie inside F(Y). This case is covered by the following definition. D~Fn,nrlO~ 4. A system (*) has the a-DS (a-decision stability) property with respect to some family ~ of fuzzy sets defined on X if the system has the a-S property with respect to ~ and maxr_~r)/~u.R(y)< max~r)#u.R(y ). E.g. the fuzzy sets U and W of Example 5 can be treated as belonging to A. Now we shall prove necessary and sufficient conditions for a system to have the a-DS property with respect to some ~. It is assumed that R f f i ( A ~ BOor... or(A,~ B,), Aj nonfuzzy.

ON S T A B I L I T Y O F F O R M A L F U Z Z I N E S S SYSTEMS

65

TI-~OREM 9. Let the system (*) hat~ the a-S property with respect to the family A. Then this system has the a - D S property with respect to the same ~ if and only if there is some p E { 1..... s} such that: (a) Ita,(y)
Co) max,m~,,(y) >~, (c) maximA,/~u(x) >a. Proof. Necessity:

V- t~u.,(yj)= j~l

V ~,u(~)A~,~,(yj)V'" v V ~v(x)/X~.(.vj) ) jll

xEA I

xEAs

=

V j-- I

x~A

v I

V j-- 1

xEA a

First, we must exclude the case when for each p ffi 1..... s, #a,(Y)• 0 for some non-feasible y, because on the strength of a-S property, for each p we have V xEA, tXv(X) < a, and the VT- i I~v.R(Y:) < 0, contradicting the a-DS property. There is then some p E { l ..... m} such that/~m,(y)--0 for any nonfeasible y, and by Definition 4, V y E r ( r ) # v . s ( y ) > a • V y ~ r - ~ r ) # v . s ( Y ) . But this means that V~.llttloR(yj)-~Vy.l#a,(yj)AVxea,#tl(X)>a. Then I t ~ ( y ) > a for so.me feasible y [y E F(Y)], and ltu(x) > a for some x CAp. The proof of the converse implication is omitted on account of its simplicity. • Theorem 9 as well as Theorem 8 can be interpreted as an operational tool for the description of a family d~ in question, or as a tool for the separation of inputs U belonging to A from those not belonging there. This interpretation is supplied by the following two corollaries.

COROLLARY 1. I f for each p -- 1..... s we have tts,(y) • 0 for some y E Y F(Y), then A = 0 . This flows from the proof of necessity. E X A M P L E 6. Let X, Y , A , B , C be as in Example 5, but F ( Y ) = {Y~,Y2,Ys} with a = 0.4. The nonfeasible set equals Y - F(Y)-- { Y3,Y4}, and

~B(y3) • 0.4,

~n(y,)>0.4 ~

V ~v(x)=~:(xO
~c(ys)>0.4

~

~/ xEl~onA

~v(x)
66

ALEKSANDER KANIA ET AL.

These implications hold because of the a-S property which must be satisfied. But if V x ltu(x)< 0.4, then m

V

V

y~ Y

(t,,(yj)/x V

j--IX

V

xEA

xEnonA

= ~X/tta(Y.j)AO.4Vp.c(Y./)AO.4==0.4, j-i

which contradicts the definition of the a-DS property. This means that there is no fuzzy input which would satisfy the definitional requirements. COROLLARY 2. I f there is some p ~ {1 ..... s} such that V~y)bt~,(y)>cx and V::~ r - ~r)P~, ( Y ) < a, then ~ v~ O and • contains any fuzzy sets U fulfilling the following conditions: (a) V ~ A , Fu(X) < 0 if l~B,(Y) >a for some y E Y - F ( Y ) and k, and Co) V~EA~/~X)>a for some l such that l~a~(y)". EXAMPLE 7. In Example 6 let F ( Y ) = {Yz,Y2,Y3}, and as before let

B=[0

0.2

0.4

0.7

1],

C--[I

0.8

0.6

0.3

0].

The fuzzy set B satisfies the premise of implication (a), namely ~tB(y4)= 0.7 > a =0.4. Then Fu(xl) ----0.4. But the fuzzy set C is adequate for the case (b) because /~c(y4)=0.30.4 for i--1,2,3. Then /~(xl) ought to be ~eater than a for i = 2, 3,4. Therefore, for the given case the family ~ can be defined as follows: ~={U:~tu(Xt)<0.4, ~tu(xl)>0.4 for i=2,3,4}. E.g. the fuzzy set U--J0.2 0.5 0.7 0.9] gives output UoR=[0.9 0.8 0.6 0.3 0.2], which remains in conformity with Definition 4. 6.

a-fl STRONG DECISION STABILITY (a-fl SDS)

This kind of stability is a kind of a-DS with an additional condition: that the maximum of the membership function inside the feasible subset of the output argument must exceed the given value p. Formally this can be written as follows: DEFn~ON 5. A formal fuzziness system (*) has the a-~ SDS property with respect to some family tg fff (a) the system (*) has the a-DS property with respect to t~, and (b) VyE~,/tU.R(y);~/3 > a .

ON STABILITY OF FORMAL FUZZINESS SYSTEMS

67

As before, we give necessary and sufficient conditions for the system (*) to have the a-fl SDS property with respect to some family ~. We assume that R-- (A i ~ B 0 o r . . , or(A,~B,), and the Ai are nonfuzzy. THEOREM 10. A system (*) has family A of fuzzy sets defined on X with respect to ~, and Co) there is y~Y-F(Y), and for any U ~ A

the a-fl SDS property with respect to some if and only if (a) the system (*) has a-DS some p E { l ..... s} such that /tB,(y)fl,

V. ~Ap~u(x) > #. The proof can be constructed analogously to that given for the Theorem 9. Thus we omit it. This parallelism of proof is due to the sufficiency of a-fl SDS for a-DS. The following corollary gives an operational rule of description of the family ~ under which the system (*) has the ~-fl SDS property. COROLLARY 3. Provided that A v~O, any U E A is characterized by the following two conditions: (a) I f there is some y E Y - F ( Y ) such that pB,(y)>a, then /~o(x)<0 for every x E.4k; (b ) V x e A, #u( X ) > fl for some p such that ttn, ( y ) < a for any y E Y - F( Y ) and The decision procedure described by Corollary 3 can be investigated in the following example.

.,Uu,q

~o

tO'

08

0.8.

O.6

0.6-

O.zr o.2.

02 t

2

3

/~

:~

X--

Fig. 4

68

A L E K S A N D E R K A N I A ET AL.

E X A M P L E 8. Let us consider the case presented in Example 5 with the additional condition fl ffi 0.6. Then, as was confirmed, the inequality Ps(Y+)> 0.7 implies the restriction V x e A l ~ u ( X ) f f i # u ( x O < 0 . 3 [See Corollary 3, point (a)]. Besides,/~c(Y4) < a,/Jc(Ys) < a, and/Jc(Y~) ;~ 0.6 for i ==1,2, 3; this implies, according to Corollary 3, point (b), that Vx~nonA t t ~ x ) = f l , i.e. /tu(x~),fl for some i ~ {2,3,4}. For instance, let U=[0.2 0.6 0.6 0.3]. This U fulfills the above condition and belongs to the family A. In fact, U o R = [ 0 . 8 0.8 0.6 0.3 0.2]. As can be seen in Fig. 4(b), / ~ u ( x 0 = 0 . 2 < 0 . 3 and Vx~nn~AI~u(x)=0.8. A s a result, V ~ E ~ r ) l ~ u . R ( x ) > O . 6 = f l , as Fig. 4(b) shows. Therefore, according to the conditions given in this example, the family A = { U : / ~ u ( x l ) < 0 . 3 , V~.~A ~u(x) > 0.6}. 7.

CONCLUSIONS

In practical applications of fuzzy-system theory the problem of stability emerges. This problem is treated in this paper in few mutations, corresponding to possibly more or less rigorous practical needs. As can be seen, the tools of analysis are oriented toward the algorithmic approach. This enables one to solve the following general problem: Given a relation matrix R and a nonfeasible set Y - F(Y), describe the family of inputs A under which system (*) has the stability property in the sense of a-S, a-DS, and a-fl SDS. REFERENCES 1. R. E. Bellmann and L. A. Zadeh, ~ort-making in fuzzy environment, Management $ci. 17, No. 4, (Dec. 1970). 2. W. J. M. Kickert and E. H. Mamdani, Analysis of fuzzy logic controller, Fuzzy Sets and @stems 1:29--44 (1978). 3. E. H. Mamdani, Advances in the linguistic synthesis of fuzzy controllers, lnternat. J. Man-Machine StudieJ 8:663-678. (1976). 4. C. V. Negoita and D. A. Ralescu, Applications of Fuzzy Sets to @stem Analysis, B ~ , Basel, Stuttgart, 1975. 5. IL M. Ton& Analysis and control of fuzzy system using finite discrete relations, Internat. J. Contro/27(3):431--440 (Mar. 1978). 6. L. A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. @stems, Man, and C'~bernet. SMC-3 (1):28-44 (Jan. 1973). Received February1980