specification and optimization of fuzzy systems using convolution techniques

Signal Processing 80 (2000) 935}949

Speci"cation and optimization of fuzzy systems usingconvolution techniques

Felipe FernaH ndez*, Julio GutieH rrez

Dep. Tecnologn&a Foto& nica, Facultad de Informa& tica U.P.M. Campus de Montegancedo, 28660 Madrid, Spain

Received 30 April 1999; received in revised form 15 November 1999

Abstract

Fuzzy logic is now widely accepted as a formal tool for describing control or decision-making processes that are basedon incomplete, vague or uncertain information. This paper presents a new approach for the description and e$cientcomputation of fuzzy rules based on fuzzy logic and also on convolution techniques. The fuzzy rules MR

rN of the MISO

considered are an extension of the zero-order Takagi}Sugeno fuzzy model and are given in the form of Rr: If X

1is

Ar1

and 2 and XN

is ArN

then z is cr, where X

jare fuzzi"ed input variables, A

rjare fuzzy numbers which belong to the

corresponding partition of unity MArjN and c

ris a nonfuzzy singleton term of output variable z. The global consideration

of the fuzziness (imprecision) of each input Xjand the fuzziness (vagueness) of the corresponding fuzzy partition MA

rjN

greatly simpli"es the corresponding speci"cation process and the involved matching computation. Two general fuzzypartitions are introduced: F-splines and /-splines, both capture the fuzziness of input variables and fuzzy termsconsidered, and the smoothness constraints of outputs. As an application example, the control of an inverted pendulum isanalyzed. ( 2000 Elsevier Science B.V. All rights reserved.

Zusammenfassung

Die Fuzzy Logik ist mittlerweile als formales Hilfsmittel zur Beschreibung von Regelungs- odcr Entscheidungsprozes-sen weit akzeptiert, wenn nur unvollstaK ndige, unklare oder unsichere Informationen vorliegen. Dieser Artikel stellt einenneuen Ansatz zur Beschreinbung und e$zienten Berechnung von Fuzzy Regeln vor, der auf Fuzzy Logik und dermathematischen Faltung basiert. Die betrachteten Fuzzy Regeln MR

rN des MISO-Systems stellen eine Erweiterung des

Takagi}Sugeno Fuzzy-Modells nullter Ordnung dar und werden in folgender From angegeben. Rr: Wenn X

lA

rlist und

2 und XN

ArN

ist, dann ist z cr, wobei die X

jdurch Fuzzy-Versionen der Eingabevariablen, A

rjdurch Fuzzy-Zahlen,

die zur entsprechenden Zerlegung der Einheit MArjN gehoK ren und c

rdurch einen einelementigen, nicht-fuzzy Term der

Ausgabevariablen z gegeben sind. Die globale Betrachtung des Fuzzy-Grades (Ungenauigkeit) jedes Eingangs Xjund des

Fuzzy-Grades (Verschwommenheit) der entsprechenden Fuzzy-Zerlegung MArjN erleichtert den zugehoK rigen

Spezi"kationsprozess und die damit verbundene UG bereinstimmungsberechnung erheblich. Es werden zwei allgemeineFuzzy-Zerlegungen eingefuK hrt: F-Splines und /-Splines. Beide erfassen den Fuzzy-Grad der Eingabevariablen und derbetrachteten Fuzzy-Terme, sowie die Glattheitsbedingungen der Ausgaben. Als Anwendungsbeispiel wird die Regelungeines invertierten Pendels analysiert. ( 2000 Elsevier Science B.V. All rights reserved.

Re2 sume2

La logique #oue est a preH sent largement accepteH e comme outil formel pour deH crire des processus de contro( le et de prisede deH cision qui reposent sur des informations incompletes, vagues ou incertaines. Cet article preH sente une nouvelle

*Corresponding author.E-mail addresses: [email protected] (F. FernaH ndez), jgr@dtf.".upm.es (J. GutieH rrez)

0165-1684/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 5 - 1 6 8 4 ( 0 0 ) 0 0 0 1 2 - 8

approche pour deH crire et calculer e$cacement des regles #oues baseH es sur de la logique #oue et aussi sur des techniquesde convolution. Les regles #oues MR

rN du MISO consideH reH sont une extension du modele #ou d'ordre zeH ro de

Takagi}Sugeno et sont sonneH es sous la forme de Rr: si X

jest A

rlet 2 et X

Nest A

rN, alors z est c

r, ou les X

jsont des

variables d'entreH e rendues #oues, les Arj

sont des nombres #ous qui appartiennet a la partition correspondante de l'uniteHMA

rjN et c

rest un terme singleton non #ou de la variable de sortie z. La consideH ration globale du caractere #ou (de

l'impreH cision) de chaque entreH e Xj

et le caractere #ou (vague) de la partition #oue correspondante MArjN simpli"ent

grandement le processus de speH ci"cation correspondant et le caractere #ou des variables d'entreH es et des termes #ousconsideH reH s, ainsi que les contraintes de douceur des sorties. Comme exemple d'application, nous analysons le contro( led'un pendule inverseH . ( 2000 Elsevier Science B.V. All rights reserved.

Keywords: Takagi}Sugeno controller; Fuzzy partition; Spline; Cross-correlation/convolution; Fuzziness; Smoothness

1. Introduction

The last few years we have seen increasing inter-est in fuzzy systems research and applications. Thisis mainly due to the success of fuzzy technology inmany "elds of engineering including consumerproducts, transportation, manufacturing, medical,control and signal processing systems.

Classical models try to avoid vague, imprecise oruncertain information because it is considered ashaving a negative in#uence in the correspondingtechnique. Fuzzy systems on the other hand delib-erately make use of this kind of information. Thisusually leads to simpler and more suitable models,which are easier to handle and are more familiar tohuman thinking.

Fuzzy engineering is mainly based on if}thenrules. These rules must not be understood in termsof logical implications, but in terms of a functionde"ned on a number of fuzzy regions. The anteced-ent of a rule basically consists of a fuzzy descrip-tion of the corresponding function domain.Additive fuzzy systems average the correspondingoverlapping rules. This description providesa systematic procedure of transforming a rule-based system into a nonlinear mapping and it is atthe present time strongly related to function ap-proximation theory. Fuzzy systems can be con-sidered included in general system theory. In thepast fuzzy systems have been strongly based onheuristic knowledge of control. This may producea misleading impression that the relationship be-tween classical and fuzzy systems is very limited.Those may also lead to the erroneous conclusionthat most of the tools of linear time-invariant

system theory completely lose their validity in thefuzzy domain.

The approach of this paper is mainly based onthe additional application of convolution tech-niques and theory of spline functions [3,4,6,8,14] tocharacterize the fuzziness of a fuzzy system. Thisassociation of fuzzy and convolution concepts givesa powerful framework to derive interesting proper-ties for the description, optimization and imple-mentation of the corresponding fuzzy rules.

The main speci"c di!erential characteristic ofthis approach from other similar approaches[1,16,20,21] is the inclusion of a general fuzzi"ca-tion function (convolution operator). This operatormakes easy the description of the input variablesimprecision and the global fuzziness of the corre-sponding input linguistic terms, and also improvesthe output smoothness of the corresponding fuzzycontroller. This fuzzi"cation function [7,11] trans-forms the observed inputs (nonfuzzy singletons x)into symmetrical fuzzy variables (even signal-fuzzynumber X), or is alternatively applied, in this paper,to the original linguistic terms considered (fuzzypartitions MA

jN). A standard inner product is used

to characterize the corresponding matching process(X is A

j) instead of the most common max}min

matching [5,9,11,18].However, in most industrial software or hard-

ware fuzzy tools, input variables are not fuzzi"ed,and singleton fuzzi"cation [7] is generally usedin the speci"cation and implementation of fuzzysystems. The main reasons given for this use areordinarily two: the general acceptation that it isnot well justi"ed in practice the need of otherfuzzi"cation functions, and the fact that general

936 F. Ferna& ndez, J. Gutie& rrez / Signal Processing 80 (2000) 935}949

fuzzi"cation functions add computational com-plexity to the direct implementation of the corre-sponding inference process.

In this paper, we intend to show that bothreasons are only partially true. First, we can addexpressive power to the corresponding fuzzysource language, including the possibility ofmore general fuzzi"cation functions. These func-tions can help to globally capture the fuzziness ofthe system in an easy and direct way: imprecisionof inputs, vagueness of antecedent terms, and alsodesired smoothness of outputs. Second, we candramatically reduced the complexity of the corre-sponding matching process, by transforming (com-piling) the original set of linguistic labels into a newfuzzier one, applying the fuzzi"cation function notto the inputs but to the corresponding antecedentlinguistic terms.

From a theoretical point of view our approachprovides a formal method for the global considera-tion of fuzziness of input variables and commonfuzziness of linguistic terms of fuzzy propositions.This way, a fuzzy proposition (X is A) with twocomponents (X, A) can be transformed intoa three-component system (dx, +A, U): a singletoninput variable (d

x), a crisp linguistic term (+A) and

an additional signal-fuzzy number (U) which glo-bally captures the fuzziness of X and A.

The rest of the paper is organized as follows:Section 2 reviews some related fuzzy basic conceptsand introduces some new ones; Section 3 summar-izes the main characteristics of the fuzzy controllermodel used; Section 4 describes the main propertiesof convolution operation on fuzzy partitions; Sec-tion 5 presents F-splines that are de"ned to modelthe uniform fuzziness of a partition; Section 6 pres-ents u-splines that are specially adapted to modelthe nonuniform and uniform fuzziness of a fuzzypartition; Section 7 shows the control of an in-verted pendulum as an application example; someconclusions are drawn in Section 8.

2. Fuzzy functional concepts

The basis of fuzzy set theory is the notion of fuzzyset. Many authors found di!erent ways of de"ningfuzzy sets [10}13]. In this paper we consider the

following functional de"nition: a fuzzy set A isa function A:;P[0,1], where; is the correspond-ing discrete or continuous domain. Such a functionis called a membership function.

In this section, we recall some de"nitions andconcepts of discrete and continuous fuzzy sets[1,10}12], for clarifying the notation used and forconvenience of the ulterior development. For a gen-eral fuzzy set A de"ned on the universe ;, thefollowing standard de"nitions are considered.f Support of fuzzy set A is a crisp set de"ned by

supp(A)"MuDA(u)'0N. (1)

f Core of fuzzy set A is a crisp set de"ned by

core(A)"MuDA(u)"1N. (2)

f Height of fuzzy set A is a crisp set de"ned by

height(A)"supu|U

A(u). (3)

A fuzzy set A is normal when height(A)"1 andits subnormal when 0)height(A)(1. For in-terest of ulterior development, we introduce thenonnormal class to designate nonstandard fuzzysets with 0)height(A)(R.

f Kernel of fuzzy set A is in this paper de"ned bythe following crisp set:

kernel(A)"MuDA(u)"height(A)N (4)

f Cardinality of fuzzy set A, on a discrete domain;, is de"ned by

DAD" +u|U

A(u). (5)

f Area of fuzzy set A, on a continuous domain;, isde"ned by

DAD"PU

A(u) du. (6)

f Any set that is not crisp has some degree offuzziness that results from the imprecision of itsboundaries [11,9]. For measuring fuzziness ofa fuzzy set, we can use the function f to expressthe lack of distinction between the membershipA(u) and its complement 1!A(u) [11,9]

f (A)"PS611(A)(1!2DA(u)!0.5D) du. (7)

F. Ferna& ndez, J. Gutie& rrez / Signal Processing 80 (2000) 935}949 937

Fig. 1. Nonnormal bounded % fuzzy number A.

f The concept of convex normal fuzzy number[11,9] captures our intuitive conception ofa fuzzy set which is around a given real number(K fuzzy number or ¸}R fuzzy number) oraround a given real interval of real numbers (%fuzzy number or ¸}R fuzzy interval). Member-ship functions A( ) ) of normal % fuzzy numbers(Fig. 1) that conform to this intuitive conceptioncan be expressed in a general form, using a condi-tional case notation [2]

A(u)"A(u;a, b,a,b,c;¸(u),R(u))

"G(u)a!asu*b#b)P 0,

(a)u)b) P c,

(a!a(u(a) P c )¸((a!u)/a),

(c(u(b#b) P c )R(u!b)/b)

(8)

where parameters (a, b, a, b, c) are real numbers:a)b, a*0, b*0, c"height(A)"1, and ¸( ) )and R( ) ) are monotonically decreasing functionsde"ned from R` to [0,1], with ¸(0)"R(0)"1.In this paper and in most practical cases,¸( ) ) and R( ) ) are one-to-one continuous func-tions. Parameters a and b determine thecore of A: core(A)"[a, b]. Parameters a and bare, respectively, called left and right spreads:¸spread(A)"a, Rspread(A)"b and Spread(A)"a#b. The spreads determine the length of supp(A): DDsupp(A)DD"(b#b)!(a!a) and in#uencethe fuzziness of A. Membership functions A( ) ) ofbounded " fuzzy numbers also conform thesede"nitions but a"b.

f Nonnormal or subnormal fuzzy numbers A(S-fuzzy numbers) with height(A)"c and (0)c(R) or (0)c(1) respectively, are introduc-ed in this paper as an extension of normal ones(Fig. 1). These new fuzzy numbers provide a newtool for generalizing and transforming standardfuzzy numbers.Usually, a set of linguistic terms is used in the

antecedent description of a fuzzy controller. Theselinguistic terms can be seen as qualitative valuesused to describe fuzzy rules. The set of M standardlinguistic terms MA

1, A

22A

MN with (0)A

j)1)

is de"ned in the domain; of a given scalar variableu. It is generally required that the linguistic termssatisfy the roperty of coverage, i.e. for each domainelement is assigned at least one fuzzy set A

jwith

nonzero membership degree.For practical reasons, the linguistic terms are

usually constrained to be normal or subnormalfuzzy numbers. Also the model here considered,and presently in many other fuzzy control models[1,11,12,16], imposes stronger conditions on thelinguistic terms used:f Partition of unity MA

jN of a domain ; is de"ned

by the following constraints:

M+j/1

Aj(u)"1, ∀u3;, (9)

A +u|U

Aj(u)B'0, 1)j)M. (10)

Eq. (9) means that for each u, the sum of all mem-bership degrees A

j(u) equals one, and Eq. (10) im-

plies that none of the fuzzy sets Aj(u) is empty. This

fuzzy segmentation of domain; is also called fuzzypartition [1,11]. A hard or crisp partition M+A

jN

can be considered as a particular case: +Aj(u)30,1.

For discrete domains ;"Mu3ZD(1)u)Q)N afuzzy partition can be conveniently represented inmatrix notation: [A]"[A

j, u] where [A] is an

M]Q-matrix whose rows [Aj] de"ne point wise

the membership functions of the correspondingfuzzy terms.

A sequence ¹"M(t0, t

1,..., t

M)D(a"t

0(t

1(2(t

M"b)N de"nes two basic partitions of

unity:


Fig. 2. Basic fuzzy partitions: (a) Crisp partition. (b) Triangular fuzzy partition.

f A crisp partition of [a, b]3; determined by a setof M crisp intervals M+A

0,+A

1,2,+A

M~1N"

M[t0, t

1), [t

1, t

2),2,[t

M~1, t

M)N (Fig. 2a).

f A triangular partition of [a, b]3; determinedby a set of M#1 triangular fuzzy numbersMA

0, A

1,2, A

MN composed of piecewise linear

triangular functions MAjN, with A

j(tj)"1 and

Aj(tk)"0 if kOj. (Fig. 2b).

In this paper, a particular extension of a fuzzypartition is considered in order to avoid boundproblems in window-convolution operations ap-plied on fuzzy partitions, which are used to modelfuzzi"cation transformations [9]:f A constant extension of a fuzzy partition

MAjN` (1)j)M) de"ned on a domain ;"

[a,b] onto a bigger domain ;`"[a!w, b#w]D(w'0) is characterized by

MAj(u)N`"G

u3[a, b] P MAj(u)N,

u3(a!w, a) P MAj(a)N,

u3(b, b#w)P MAj(b)N

(11)

f An important parameter of a fuzzy partition isthe overlapping factor ov, which is the maximumnumber of membership functions with nonnullmembership degrees for each element u of do-main ;.

ov(MAjN)"max

uA+

j

vAj(u)wBu3;, 1)j)M,

(12)

where vaw denotes the ceiling function (smallestinteger *a).We also introduce in this section, a speci"c type

of nonnormal fuzzy number that we use as a

conceptual bridge between fuzzy and signal process-ing systems, to accomplish some standard signaloperations. The [0,1] membership boundednesscondition is relaxed to allow general convolution}correlation operations on fuzzy sets.f A c-normal S-fuzzy number X(u) (nonnormal

fuzzy number with cardinality unity) is de"ned inthis paper as

X(u)"1

D>D)>(u), (13)

where >(u) is the membership function of a stan-dard nonempty fuzzy number> de"ned in a dis-crete or continuous domain ;, and D>D is thescalar cardinality or area of >. Therefore, X(u) isa unit-area S-fuzzy number. It is convenient tonotice, that unit impulse functions (for discretedomains)

dx"d(u!x)"M(u"x)P1;0N (14)

or informally Dirac pulses (for continuous do-mains) can be included into this type of S-fuzzynumbers.Finally, in this section we introduce the concept

of fuzzixcation operator, which is a generalization ofthe concept of fuzzi"cation function [11,18]. Thisoperator allows a uni"ed consideration of the fuzzi-ness associated to the input variables or to thecorresponding antecedent terms of a fuzzy system.f The fuzzi"cation operator applies an even fuz-

zi"cation function U (even c-normal S-fuzzynumber) onto a fuzzy number C (normal or sub-normal), de"ned on a continuous extended do-main;, and gives a new fuzzy number C@ de"ned


by means of the following cross-correlation/con-volution operation

UHC(x)"PU

C(u) )U(u!x) du

"SC(u),U(u!x)T"C@(x). (15)

A Particular interesting case of fuzzi"cation isthe singleton fuzzi"cation operator d:

dHC"C. (16)

It is convenient to note that this fuzzi"cationconvolution modi"es, in general, the support andcore of the corresponding fuzzy number:

DDsupp(UHC)DD"DDsupp(U)DD#DDsupp(C)DD, (17)

DDcore(UHC)DD"max(0,DDcore(C)DD!DDsupp(U)DD),

(18)

where DD ) DD stands for the length of the corre-sponding crisp interval. The kernel of this convo-lution product also veri"es the followingsymmetrical expression:

DDkernel(UHC)DD

"max(0, DDkernel(C)DD!DDsupp(U)DD,

DDkernel(U)DD!DDsupp(C)DD). (19)

3. Fuzzy system model

In this section we summarize the main character-istics of the fuzzy implementation model con-sidered. It is assumed that the fuzzy system hasN fuzzi"ed inputs: (X

1, X

2, 2 X

N), de"ned within

a universal set ;N, and one nonfuzzy-singletonoutput z, de"ned within a universal set <.

This MISO fuzzy system is a generalized zero-order Takagi}Sugeno (T}S) fuzzy system [1,7] andit is de"ned by a set of rules MR

rN, which are given in

a canonical conjunctive normal form

Rr: If(X

1is A

r1and 2 and X

NisA

rN)

then (z is cr), (20)

where Xi

is a fuzzi"ed input variable (an evenc-normal S-fuzzy number), A

r,iis a normal fuzzy

number which belongs to a corresponding partitionof unity MA

rjN and c

ris a nonfuzzy singleton term

associated to output variable z of the rth rule.Before describing the implementation model

proposed, it is convenient to note that the smooth-ness of the output function of a fuzzy system MR

rN

directly depends on the smoothness of antecedentmembership functions (X and A). This fact restrictsthe choice of membership functions that we can use.For instance, the frequently used trapezoidal mem-bership functions for linguistic terms MA

jN, and

singletons for inputs X, result in a nonsmoothoutput.

The evaluation of output z of the T}S fuzzycontroller referred, with N fuzzy number input vari-ables X

ion a domain ;, and M rules R

r, is deter-

mined via a sequence of a three-stage procedure:f Matching. For a discrete domain;, the standard

procedure computes, for each fuzzi"ed inputvariable X

iand each fuzzy number term A

ri, the

degree of cross correlation uri

(0)uri)1):

uri" S

u|U

(¹(Xi(u),A

ri(u)))"+

u|U

(Xi(u) )A

ri(u)),

(21)

where ¹ and S mean t- and s-norm, respectively[11]. In this paper, for the partitions of unityMA

jN and the c-normal S-fuzzy numbers X

icon-

sidered, ¹ is implemented by the algebraic prod-uct (which is a di!erentiable function) and S bythe algebraic addition (which is also a di!erenti-able function).Note that we have substituted the standard

min}max matching operation [5,9,11,18] witha product}sum matching operation. A comparativeadvantage of this product}sum matching operation(more formally described in Theorem 1 of nextsection) is that it is a partition of unity preservingtransformation. Unfortunately, this property is notaccomplished by min}max matching operationusually considered [9].

For a continuous domain ; the following ana-logous expression holds

uri"P

U

¹(Xi(u),A

ri(u)) du"P

U

Xi(u) )A

ri(u) du.

(22)


In our approach, this matching operation is pre-viously transformed (during compilation time offuzzy rules) into a standard singleton computation[9]. This transformation is accomplished by ap-plying input fuzzi"cation operator U, not to theoriginal singleton input, but to fuzzy terms of thecorresponding fuzzy partition U : MA

ijNPMA@

ijN

(this point is more deeply described in Section 4).The new obtained partition MA@

ijN is also a partition

of unity (or fuzzy partition). Furthermore, the newsingleton computation only is accomplished foractive rules (with antecedent truth degree '0). Ifov is the new overlapping factor of fuzzy partitionsMA@

ijN then the maximum number of active rules is

equal to ovN [7].Antecedent conjunction. For each active rule

r compute its activation degree by

ur"¹

i|N

(uri)"<

i|N

(uri), (23)

where t-norm ¹ is implemented by the algebraicproduct. This t-norm has also been implemented bythe min operation [11].

Rule aggregation. The output z is computed asfollows:

z"+

r|R !#5*7%(u

r) c

r)

+r|R !#5*7%(ur )

" +r|R !#5*7%

(ur) c

r), (24)

where R!#5*7%

is the set of active rules. Last expres-sion does not use any division since in a fuzzypartition the denominator of previous fractionequals one for every point of domain ;N [1,12].

4. Fuzzy convolution

Unconditional fuzzy propositions are sentenceswith truth value in the interval [0,1]. They cangenerally be expressed in a canonical form by

p : X is A. (25)

The degree of true of a fuzzy proposition can beinterpreted as a fuzzy matching (21) or (22). Thiscorrelation operation gives the degree of truth(matching) between an input variable X and a fuzzylinguistic term A [11]. The truth degree (Tr) of thisfuzzy proposition, on a continuous domain ;, can

be computed by the standard inner product [15]

Tr(X is A)"SX(u),A(u)T"PU

X(u) )A(u) du (26)

where X(u) is a c-normal S-fuzzy number and A(u)is a normal or subnormal fuzzy number.

Variable X can be considered as a fuzzi"ed inputvariable obtained by a fuzzi"cation function [11].It can be expressed by a cross-correlation/convolu-tion of an even unit-area fuzzi"cation functionUX(.) (even c-normal S-fuzzy number) with a Diracsingleton d

x"d(u!x), in order to consider the

imprecision related to x:

X(u)"PU

dx(v) )UX(u!v) dv"d

x * UX

"UX(u!x), (27)

where * stands for the corresponding cross-correla-tion/convolution operation. The fuzzy number X isconsequently the result of a convolution ona singleton input d

xto express the associated im-

precision.Therefore, the truth degree of fuzzy proposition

p, can be computed by means of the cross-correla-tion operation of fuzzy set A( ) ) and the c-normalS-fuzzy number UX(.):

Tr(X is A)"PU

UX(u!x) )A(u) du

"PU

UX(u!x) )A(u) du

"UXHA(x)"A@(x). (28)

The cross-correlation integral is similar to convolu-tion integral except that the fuzzy set X(u) is simplyshifted to X(u!x) without reversal. For the caseconsidered here, where X(.) is an even S-fuzzy num-ber, convolution and correlation are equivalent.This follows because an even fuzzy set and its imageare identical. With this restriction, convolution andcorrelation are loosely interpreted as identical.

This cross-correlation/convolution of an evenc-normal S-fuzzy number UX( ) ) on a set of linguis-tic terms of a fuzzy partition MA

jN determines a new

set of linguistic terms MA@jN; this new set of linguistic


Fig. 3. Equivalent fuzzy matching computations: (a) Tr(X is MAN), (b) Tr(dx

is MUX*AN).

terms includes the fuzziness of fuzzy variable X andof linguistic terms MA

jN.

An example of this partition transformation isshown in Fig. 3. The main advantage of this trans-formational method is that it only carries outa standard singleton fuzzi"cation in execution time,but has the same expressive power of a generalinput fuzzi"cation [11].

The following theorems show some interestingproperties of the fuzzi"cation convolution con-sidered.

Theorem 1. The partition of unity condition of a fuzzypartition MA

jN is preserved under the cross-correla-

tion/convolution

MAjN*U"MA

j*UN"MA@

jN, (29)

where U is an even c-normal S-fuzzy number ( fuzzix-cation operator).

Proof. It is a consequence of distributive propertyof convolution operation:

M+j/1

A@j(u)"

M+j/1

(Aj(u)*U)"

M+j/1

(Aj(u))*U"1*U

"1, ∀u3;. h (30)

Theorem 2. The area or cardinality of each fuzzynumber of a fuzzy partition MA

jN is preserved under

the cross-correlation/convolution:

DAj*

UD"DAjD, (31)

where U is an even c-normal S-fuzzy number (fuzzix-cation operator).

Proof. It is a consequence of commutative anddistributive property of integral operator

DAj*

UD"DU*AjD"P

UAP

U

UX(x!u) dxBA(u) du

"PU

1 )A(u) du"DAjD. h (32)

Theorem 3. Nonnormal fuzzy numbers are closedunder fuzzixcation convolution, i. e. if A

jis a nonnor-

mal fuzzy number, then Aj*

U is also a nonnormalfuzzy number, where U is an even c-normal S-fuzzynumber (fuzzixcation operator).

Proof. Since the fuzzi"cation convolution(weighted window averaging or lowpass "lter) con-sidered preserves the convexity property of a fuzzynumber A

j. h

Theorem 4. Smoothness is additive under fuzzi-xcation convolution operation: If A

jand U have

smoothness of order m and n respectively(A3Cm~2 and U3Cn~2), then A

j*U has smoothness

of order m#n (Aj*

U3Cm`n~2), where A is anormal or subnormal fuzzy number, U is an evenc-normal S-fuzzy number. A function has kth smooth-ness when its derivative of order k becomes impulsive.The corresponding diwerentiability class Ck~2 is thespace of functions that are k!2 times continuouslydiwerentiable.

Proof. Since the order of Fourier transformsF [3,17] of A

j, U and A

j*U are, respectively,

F(Aj)"o(DsD~m); F(U)"o(DsD~n);

F(Aj*

U)"o(DsD~(m`n)), (33)


Fig. 4. Equivalent fuzzy matching speci"cations: (a) Tr(X is MAN), (b) Tr(dx*

UX*UA is M+AN).

hence the smoothness order of Aj*

U is m#n (its(m#n)th derivative is impulsive) and consequentlyA

j*U belongs to the class Cm`n~2. h

Therefore, the convolution of a normal o subnor-mal fuzzy number A with an even c-normal S-fuzzynumber is a smoothing function that producesa window averaging of A. This local averaging hasthe e!ect of suppressing the high-frequency vari-ations while preserving the basic shape of A.

Last theorem has practical signi"cance in fuzzycontrol applications, since we can increase thesmoothness or continuity order of a T}S controlleroutput function by means of an additional fuzzi"-cation convolution applied to the correspondinglinguistic fuzzy terms.

Theorem 5. Variance p2 (mean-square deviation re-ferred to the centroid of the fuzzy number) is additiveunder fuzzixcation convolution operation

p2(Aj*

U)"p2(Aj)#p2(U)"p2(A@

j). (34)

Proof. Since this property holds for general convo-lution [3]. h

This variance property of convolution has animportant conceptual interpretation in the con-sidered context. If we consider as an alternativeuncertainty measurement of A

jand U the corre-

sponding variance, the new linguistic term A@j

hasan uncertainty equal to the addition of uncertaintyof A

jand U. So A@

jglobally captures the impreci-

sion of input variables and the vagueness of linguis-tic terms.

In order to simplify the speci"cation of anteced-ent terms of fuzzy rules and for achieving themaximum decomposition in the description ofa fuzzy system, the following additional consider-ations are also of particular interest.

If all linguistic terms (fuzzy numbers) of a fuzzypartition MA

jN have the same bound shape (same

functions ¸"R, and same spreads a"b), the fuz-ziness of this fuzzy partition is uniform. In this case,it is convenient to specify the corresponding fuzzypartition MA

jN by means of a convolution operation

applied on each crisp set of the original crisp parti-tion M+A

jN. Each term of fuzzy partition A

j(.) is

obtained by the convolution with an even unit-areafuzzi"cation function UA(.) (even c-normal S-fuzzynumber) applied on crisp set +A

j(.):

Aj(u)"P

U

+Aj(.) )UA(u!v) dv"+A

j*UA(u). (35)

Consequently, the truth degree of fuzzy propositionp, can be computed by means of the multiple con-volution/cross-correlation operation

Tr(p)"Tr(X is A)"UX*A

"UX*(UA*+A)"(UX*UA)*+A"AA, (36)

where the convolution product UX*UA capturesthe uniform fuzziness of fuzzy terms and the fuzzi-ness of the corresponding input. Fig. 4b showsa fuzzy system equivalent to the one shown in Fig.4a, de"ned by a partition of unity of crisp linguisticterms and a c-normal S-fuzzy number as inputvariable. Note that we have used di!erent scales torepresent the ordinates of partition MAjN and theordinates of variable X in Figs. 3 and 4.


Fig. 5. F-spline partitions: (a) Nonuniform crisp partition MA1jN, (b) F-spline fuzzy partition MA5

j[17]N.

Next section presents a new class of fuzzy parti-tions that will be called F-spline partitions, whichare specially adapted to model the uniform fuzzi-ness of a fuzzy system.

5. F-spline partitions

The "rst step of a fuzzy controller design is thedivision of input space into fuzzy regions. Thesimplest solution is a rectangular crisp partition, i.e.to divide each input domain [a, b]3; into an or-dered set of M intervals: a"t

0(t

1(2(

tM"b, and assign each region to a crisp linguistic

term +Aidenoted in this section by A1

jde"ned by

A1j(u)"M(t

j)u(t

j`1)P1; 0N"1

*tj,tj`1)

j"0,2,M!1. (37)

A possible second step is the introduction of a uni-form fuzzi"cation function N to express the com-mon fuzziness of the corresponding system.

In one-dimensional space, the initial basisconsidered for fuzzi"cation functions is the evenrectangular c-normal S-fuzzy number (of unit-areaand base o) N1[o]:

N1[o](u)"M(DuD(o/2)P1/o; 0N"(1/o)*~o@2,o@2+.

(38)

The matching between the function N1[o] anda crisp domain partition MA

jN"MA1

jN is a new fuzzy

partition MA2j[o]N de"ned by the multiple convolu-

tion operation:

MA2j[o]N"N1[o]*MA1

jN"MN1[o]*A1

jN. (39)

The resulting trapezoidal or triangular fuzzy termsMA2

j[o]N are in general normal or subnormal fuzzy

numbers, and also form a partition of unity (fuzzypartition).

An n-order B-spline fuzzi"cation function can bede"ned by the following recurrence equation:

Nn[o]"M(n"1)PN1[o]; N[o]*Nn~1[o]N. (40)

In general, if an n-order B-spline fuzzi"cation func-tion Nn[o] is correlated with a crisp partition MA1

jN:

Nn[o]*MA1jN"MNn[o]*A1

jN"MAn`1

j[o]N, (41)

it gives a new fuzzy partition MAn`1j

[o]N which iscalled in this paper, F-spline partition of ordern#1, formed in general by normal and subnormalfuzzy numbers.

The parameters n (order) and o (rectangular win-dow size) of B-spline determine the support andshape of the corresponding convolution "lter usedand therefore the smoothness of the correspondingcontrol surface obtained. For example, in Fig. 5 it isshown a nonuniform F-spline fuzzy partitionMA5

j[17]N (obtained by successive application of

four convolutions with a rectangular window sizeo"17) derived from the corresponding nonuni-form crisp partition MA1

jN:

MA5j[17]N"N4*MA1

jN. (42)

This way the speci"cation of complex linguisticfuzzy partitions for controlling the smoothness ofa fuzzy controller, is not always necessary, since theuser can factorize the regular fuzziness of a linguis-tic fuzzy partition, and specify it as an additionalfuzziness of the corresponding input. Therefore, inregular fuzziness domains it is possible to specify, in


Theorem 6. For each n*0, F-splines An`1j

[o] of degree n and (n#1)th order have the following properties:

1. Initial support and core size: DDsupp(A1j)DD"DDcore(A1

j)DD.

2. Final support size: DDsupp(An`1j

[o])DD"DDsupp(A1j)DD#n*o.

3. Final core size: DDcore(An`1j

[o])DD"max(0,DDcore(A1j)DD!n*o).

4. Final kernel size: DDkernel(An`1j

[o])DD"max(0,DDcore(A1j)DD!n*o,M(n"1)Po!

DD supp(A1)DD;0N).5. Initial and xnal overlapping factor: ov(MA1

jN)"1 and ov(MAn`1

jN)*2 for n*1.

6. Nonincreasing height: height(An`2j

[o]))height(An`1j

[o]).7. Normality of trapeziums: height(A2

j[o])"M(DDsupp(A1

j)DD*o)P1( HnormalH);

supp(A1j[o])/o ( HsubnormalH)N.

8. Nondecreasing overlapping factor: ov(MAn`2j

[o]N)*ov(MAn`1j

[o]N).9. Continuity class: MAn`1

j[o]N3Cn~1 for n*1.

10. Symmetry: An`1j

[o] is symmetric with respect to the centre of A1j.

11. Polynomial form: An`1j

[o] are formed by piecewise polynomial functions of degree n.12. Increasing fuzziness: An`2

j[o] are fuzzier terms than An`1

j: f (An`2

j[o])'f (An`1

j[o]).

13. Final support: If A1j"1

*tj,tj`1)then supp(An`1

j[o])"[t

j!n*o/2, t

j`1#n*o/2).

14. Final border nullity: If A1j"1

*tj,tj`1)then An`1

j[o](t

j!n*o/2)"

An`1j

[o](tj`1

#n*o/2)"0 for n*1.

an equivalent form: crisp linguistic partitions andfuzzi"ed inputs, for the antecedents of the rules.

Some important properties of F-splines, relevantto the design and implementation of fuzzy control-lers are shown below.

Sketch of proof. These properties are direct conse-quence of general convolution properties [3,6,8]and fuzzi"cation Convolution Theorems 1}5 ofSection 4 applied to the particular case of nonuni-form crisp partitions. h

Property 9 has special practical interest to deter-mine the output continuity order (smoothness) ofthe T}S controller considered. Note that it is alsopossible to apply in cascade several fuzzi"cationfunctions (B-spline convolution "lters) with di!er-ent window support (o) and order (n), in order tocharacterize several fuzziness of a fuzzy system:

U"/X*/A*/Z"Nnx[ox]*NnA[o

A]*Nnz[o

z],

(43)

derived from the imprecision of input /X, uniformvagueness of antecedents linguistic terms /A andsmoothness constraints of output /Z.

Next section presents a new type of fuzzy parti-tions that will be called u-splines, which are

specially adapted to model the uniform andnonuniform fuzziness of a fuzzy system.

6. u-spline partitions

F-splines MAn`1j

N have a uniform fuzzinessthrough the domain of fuzzy partition. If an n-orderB-spline fuzzi"cation function Nn[o] is applied tothe original crisp partition MA1

jN that veri"es:

DDSupp(Nn[o])DD"n )o)minj

DDsupp(A1j)DD

j"0,2,M!1. (44)

In this case, the transformed fuzzy partition is com-posed of normal fuzzy numbers, which verify thatheight (An`1

j[o])"1 j"0,2,M!1. This condi-

tion is easy to accomplish when the interval lengths ofthe original crisp partition are approximately equal.

In many cases, especially when the length of theoriginal crisp intervals is very di!erent, it is notconvenient to apply a uniform fuzzi"cation func-tion (convolution "lter) through the whole domain.It is more advantageous to initially provide a tri-angular fuzzy partition MB

jN composed of fuzzy

numbers, with left and right fuzziness [9] directlyproportional to the corresponding lengths of inter-vals originally considered.


Theorem 7. For each n*0, u-splines Bn`2j

[o] of degree (n#1) and (n#2)th order have the followingproperties:

1. Initial core: core(B2j)"(t

j, t

j); DDcore(B2

j)DD"0.

2. Initial support: supp(B2j)"(t

j`1, t

j~1); DDsupp(B2

j)DD"(t

j`1!t

j~1).

3. Overlapping factor: ov(MB2jN)"2; ov(MBn`2

jN)'2 for n'0.

4. Initial fuzziness: f (B2j)"supp(B2

j)/2.

5. Final support size: DDsupp(Bn`2j

[o])DD"DDsupp(B2j)DD#n*o.

6. Final core and kernel size: DDcore(An`1j

[o])DD"0; DDkernel(An`1j

[o])DD"0.7. Decreasing height: height(Bn`3

j[o])(height(Bn`2

j[o]).

8. Nondecreasing overlapping factor: ov(MBn`3j

[o]N)*ov(MBn`2j

[o]N).9. Continuity class: MBn`2

j[o]N3Cn

10. Asymmetry: Bn`2j

[o] is in general an asymmetric function with respect to thecore of B2

j.

11. Polynomial form: Bn`2j

[o] are formed by piecewise polynomial functions of degreen#1.

12. Increasing fuzziness: Bn`3j

[o] are fussier terms than Bn`2j

: f (Bn`3j

[o])'f (Bn`2j

[o]).

The "rst step of the design is now the determina-tion of a set of M representative points in the inputspace. These control points should be distributedthroughout input domain [a, b]3;, and de"ne theordered set: t

0"!R(t

1"a(t

2(2(t

M"b(t

M`1"R. Many other fuzzy partitions as-

sociated to this nonuniform sample can be used,but the simplest and more common solution isa nonuniform triangular fuzzy partition, i. e. assignto each point t

j(1)j)M) a normal triangular

fuzzy number B2j

de"ned by:

B2j(u)"M(t

j~1)u(t

j)P(u!t

j~1)/(t

j!t

j~1);

(tj)u(t

j`1)P(t

j`1!u)/(t

j`1!t

j); 0N. (45)

This fuzzy partition is a nonuniform B-spline parti-tion [8] of order 2 and degree 1. The correspondingnormal K fuzzy numbers have linear shape func-tions ¸(u)"R(u)"1!u, with a

j"(t

j!t

j~1) and

bj"(t

j`1!t

j). The Fuzziness of these normal tri-

angular fuzzy number is [9]:

f (B2j)"(a

j#b

j)/2"(t

j`1!t

j~1)/2. (46)

To add a common fuzziness of the correspondingsystem, a uniform fuzzi"cation function can be in-troduced on this triangular partition of unity. Ingeneral, if an n-order B-spline fuzzi"cation functionNn[o](n*1) is convolved with a nonuniform nor-mal triangular fuzzy partition MB2

jN:

Nn[o]*MB2jN"MNn[o]*B2

jN"MBn`2

j[o]N, (47)

it gives a new fuzzy partition MBn`2j

[o]N which iscalled in this paper, u-spline partition of ordern#2, formed by normal (in the boundaries) andsubnormal fuzzy numbers.

The B-spline convolution "lter used Nn[o] alsodetermines the smoothness of the correspondingcontrol surface. For example, in Fig. 6 a nonuni-form cubic u-spline fuzzy partition MB4

j[13]N of

order n#2"4 (obtained by successive applicationof 2 convolutions with a rectangular window sizeo"13) is shown derived from the correspondingnonuniform triangular partition MB2

jN:

MB4j[13]N"N2[13]*MB2

jN. (48)

This way the designer can specify a nonuniformfuzziness by means of a normal triangular fuzzypartition, which gives a nonsmooth multilinearfunction output. Furthermore, it is possible to in-clude a uniform fuzzi"cation B-spline function toprovide a suitable smoothness for the correspond-ing output function.

Next theorem shows some main properties ofu-splines derived from the corresponding fuzzi"ca-tion process.

Sketch of proof. These properties are also directconsequence of general convolution properties[3,6,8] and fuzzi"cation convolution Theorems 1}5of Section 4 applied to nonuniform triangular fuzzypartitions. h


Fig. 6. u-Spline partitions: (a) Nonuniform triangular fuzzy partition MB2jN, (b) Nonuniform cubic fuzzy partition MB4

j[13]N.

Table 1Fuzzy rules of inverted pendulum

h

NB NS ZR PS PB

NB PB PB PB ZR ZRNS PB PS PS ZR ZR

hQ ZR PB PS ZR NS NBPS ZR ZR NS NS NBPB ZR ZR NB NB NB

7. Inverted pendulum system

As an example of nonlinear feedback controlsystem, we present now a classical experiment. Itconsists of the control of verticality of an invertedpendulum, also called cart}pole system [12]. If wecall h to the angle between the vertical and thependulum, the inputs to the controller will be h andthe angular speed hQ .

The mechanics of the inverted pendulum to-gether with its mobile platform has been simulatedusing Simulink in Matlab. The mathematicalmodel of the pendulum used for simulation hasbeen the same applied by Yamakawa [19] andDriankov et al. [7]:

I ) h$"< )¸ sin h!H )¸ cos h,

<!mg"!m¸(h$ sin h#hQ 2 cos h),(49)

H"myK#m¸(h$ cos h#hQ 2 sin h),

;!H"M ) yK .

Where 2 )¸ is the length of the pendulum, y theposition of the platform bearing the pendulum,M the mass of the platform, m the mass of thependulum, H the horizontal force at the pivot,< the vertical force at the pivot,; the driving forcegiven to the platform and I"1

3m )¸2 is the moment

of inertia. Concrete values of those parameters are:¸"0.25 m, M"0.5 Kg, m"0.6 Kg.

On the other hand, the set of fuzzy rules imple-mented in our experiment has been indicated inTable 1, which is similar to the one used by otherauthors [13].

We have started from a well-known fuzzy modelin order to depict an experiment thought indepen-dently from the techniques we have developed.

What we are trying to study is the in#uence ofthe shape of the original linguistic labels employedand the e!ect of applying convolution techniques.

Nonuniform normal triangular fuzzy partitionsare initially considered for the set of linguistic labelswith constant extension for the extremes. Bothinputs: angle and angular speed have the samelinguistic terms and the same fuzzy partition. Theuniverse of discourse is symmetrical respect to0 and it is in the range [!5,#5]. A new fuzzi"edset of labels is obtained by the application of a con-volution "lter on the original triangular linguisticterms. All the membership functions have beenconvolved two times with a c-normal rectangularpulse of 0.9 units of width or, what is the same,convolved one time with a c-normal triangle of 1.8units of base.


Fig. 7. Control surface of an inverted pendulum before applyingfuzzi"cation convolution.

Fig. 8. Control surface of an inverted pendulum after applyingfuzzi"cation convolution.

The control surface of the fuzzy controllerobtained by Takagi}Sugeno inference procedurebefore applying fuzzi"cation convolution is shownin Fig. 7 (a piecewise bilinear surface built on hy-perbolic paraboloid patches), while the "nal con-trol surface obtained after applying fuzzi"cationconvolution is shown in Fig. 8. From the compari-son of the initial C0 piecewise ruled control surfaceobtained from the triangular partition with the"nal one obtained from the corresponding C2 cubicu-spline partition, the improvement in surfacesmoothness is noticeable, which really matchesmuch well with the nature of fuzzy control. Fromthe analysis of responses supplied by the simulator,we have observed a signi"cant improvement in themovement continuousness.

8. Conclusions

In this paper we have presented an extension ofa zero-order Takagi}Zugeno fuzzy controller. Thefuzzy model developed is mainly based on the addi-tional application of standard convolution tech-niques for specifying the fuzziness (imprecision) ofinput variables, common fuzziness (vagueness) ofantecedent linguistic terms and to improve thesmoothness of output function.

Moreover, for the purpose of smoothness ande$ciency, we have also used partitions of unity forthe antecedent linguistic terms, a di!erentiablet-norm (algebraic product) and a di!erentiableaggregation operator (algebraic sum) to implementfuzzy conjunction of antecedents and disjunction ofrules, respectively.

The utilization of fuzzi"ed input variables in thedescription of fuzzy controllers gives more expres-sive power to the fuzzy source language, allowsdirect speci"cation of control-surface smoothnessand does not involve an additional computationin execution time, using the referred trans-formation.

Furthermore, we have introduced two newsplines: F-spline and u-spline. The correspondingfuzzy partitions allow de"ning a uniform andnonuniform fuzziness on linguistic terms and pro-vide a simple method to specify and implementhigh-performance fuzzy systems.

In future works, the results of the paper can begeneralized to other families of fuzzy splines.

References

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specification and optimization of fuzzy systems using convolution techniques

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