# Fuzzy Logic Controllers Are Universal Approximators

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629

IEEE TRANSACTIONS ON SYSTEMS,MAN,AND CYBERNETICS, VOL 25, NO. 4, APRIL 1995

Fuzzy Logic Controllers Are

Universal Approximators

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J. L. Castro

Abstract-In this paper, we consider a fundamental theoretical

question, Why does fuzzy control have such good performance

for a wide variety of practical problems?. We try to answer this

fundamental question by proving that for each fixed fuzzy logic

belonging to a wide class of fuzzy logics, and for each fixed type

of membership function belonging to a wide class of membership

functions, the fuzzy logic control systems using these two and any

method of d e f d c a t i o n are capable of approximating any real

continuous function on a compact set to arbitrary accuracy. On

the other hand, this result can be viewed as an existence theorem

of an optimal fuzzy logic control system for a wide variety of

problems.

1

I. INTRODUCTION

D

Fig. 1. Triangular membership function.

Manuscript received August 14, 1992; revised August 1, 1993 and June

3, 1994.

J. L. Castro is with the Department of Computer Science and Artificial

Intelligence, Universidad de Granada, 18071 Granada, Spain.

IEEE Log Number 9406630.

i.e., they are capable of approximating any real continuous

function on a compact set to arbitrary accuracy. This class is

that with:

1) Gaussian membership functions,

2) Product fuzzy conjunction,

3) Product fuzzy implication,

4) Center of area defuzzification.

Other approaches are due to Buckley [4], [5]. He has

proved that a modification of Sugeno type fuzzy controllers

are universal approximators. The modifications are:

1) The consequent part of the rules are polynomial functions, not only linear functions as in Sugeno type controllers,

2) The defuzzification is 6 = CXip(ni,m;),where X i is

the matching of the input value with the antecedent

part of the rule Ri, while in the Sugeno controller it

is X i = X;/CXj.

Although both results are very important, many real fuzzy

logic controllers do not belong to these classes. The main

reasons are that other membership functions are used, other

inference mechanisms are applied or other type of rules are

used.

The most common membership functions are the triangular

(see Fig. 1) or trapezoidal (see Fig. 2) functions. With respect

the fuzzy inference, a wide variety of fuzzy implications are

used: R-implications [21] and Mamdani implication [ 121 are

the most common. Finally, in many fuzzy controllers the

consequent part is not a polynomial function but a fuzzy

proposition or a linear or constant function.

URING the past several years, fuzzy logic control (FLC)

has been successfully applied to a wide variety of practical problems. Notable applications of that FLC systems include

the control of warm water [7], robot [6], heat exchange [15],

traffic junction [16], cement kiln [9], automobile speed [14],

automotive engineering [25], model car parking and turning

[19], [20], turning [17], power system and nuclear reactor [3],

etc . ..

It points out that fuzzy control has been effectively used in

the context of complex ill-defined processes, specially those

which can be controlled by a skilled human operator without

the knowledge of their underlying dynamics. In this sense, neural and adaptive fuzzy systems has been compared to classical

control methods by B. Kosko in [8]. There, it is remarked that

they are model-free estimators, i.e., they estimate a function

without requiring a mathematical description of how the output

functionally depends on the input; they learn from samples.

However, some people criticize fuzzy control because its

effectiveness has not been proved. That is, the very fundamental theoretical question “Why does a fuzzy rule-based system

have such good performance for a wide variety of practical

problems?’ remains unanswered. There exist some qualitative

explanations, e.g., “fuzzy rules utilize linguistic information”,

“fuzzy inference simulates human thinking procedure”, “fuzzy

rule systems capture the approximate and inexact nature of

the real world,” etc., but mathematical proofs have not been

obtained.

A first approach to answer this fundamental question in a

quantitative way was presented by Wang [ 181. He proved that

a particular class of FLC systems are universal approximators,

a

b

C

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zyxw

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0018-9472/95$04.00 0 1995 IEEE

630

zyxwvutsrqponmlkj

zyxwvu

zyxwv

zyx

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 25, NO. 4, APRIL 1995

if U is the universe of an input variable X, and X = 5 , E U

is the input value, the output of the fuzzification interface is a

fuzzy set on U , F = Fuzzy(z,). There are two main types

of fuzzification:

zyx

A) Point defuzzijication:

zyx

1 if z = x,

0 in other case

a

Fig. 2.

b

C

d

B ) Approximate defuuijkation:

Trapell iidal membenhip function.

Thus, the question: “Are these other types of fuzzy logic

controllers (which are usually applied) also universal approximators?” or, In a more general form “What other types of fuzzy

logic controllers are universal approximators?’ still remains

unanswered.

In this papcr we will answer this question for a large number

of cases. Specifically, we will prove that other classes of FLC

are also universal approximators. These classes are those with:

1) A kind of membership functions, including among others, trapezoidal or triangular membership functions,

2 ) the fuzLy conjunction modeled by an arbitrary t-norm,

3) the fuzzy implication only needs to satisfy a weak

property (R-implications and t-norms satisfy it),

4) the defuzzification method only needs to satisfy a very

weak condition (usual defuzzification methods satisfy it).

Many of examples cited in the first paragraph belong to one

of these clases. Yamakawa’s [23], Expert Systems [25],and

Sugeno’s [ 191, [20] fuzzy controllers belong to these classes.

( z - x,\ < S.

F ( z ) # 0 if and only if

There are many different kinds of fuzzy logic which may

be used in a fuzzy inference machine. The general inference

rule from a single rule

if X is A then Y is B

(2)

is the generalized modus ponens [ 131:

if X is A

X is A’

Y is B

then

Y is B‘

where

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zyxwvut

B’(y) = supT’(A’(~).I ( A ( . E )B. ( y ) ) )

X

depending on a t-norm T’ and an implication function 1.To

translate a fuzzy rule of kind (1) to a simple fuzzy rule of type

( 2 ) is used a fuzzy conjunction:

X Iis A1 and X2 is A2 and . . . X,,is A,. then y is B

ZisA

where A(.?) = T(A1(xl). A ~ ( z * ) . . . . A,(z,)) depending on

a t-norm T.

The defuzziJication interface defuzzies the fuzzy output of

a system to generate a non fuzzy output. The most commonly

used defuzzification methods are:

a ) Center of Area:

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11. FUZZYLOGICCONTROLLERS

A FLC system is composed by four principal elements: fuzzy

rule base, fiuzification interface, fuzzy inference machine,

and defuzzification interface. In this paper we consider multiinput-single-output (MISO) fuzzy logic control systems f :

U C 72” + ’R, because a multi-output system can always be

separated into a collection of single-output systems.

The fuzzy rule base is a set of linguistic statements in

the form: “IF a set of conditions are satisfied THEN a set

of consequences are inferred”, where the conditions and the

consequences are associated with fuzzy concepts (i.e. linguistic

terms). For example, in the case of a MISO FLC with n inputs,

the fuzzy rule base may consist of the following rules:

Rj:

If:rlisAiarid . . . aridx,isAA. t h e n y i s B J

(1)

( i =: 1 . . . U ) are the inputs to the fuzzy rule system,

where

y is the output of the system, A i and BJ (j= 1 . . . k )

are the linguistic terms, and k is the number of fuzzy rules.

By relating each linguistic term in the fuzzy rules with a

membership function, we specify the meaning of the rules.

There are many different kinds of fuzzy rules: see [lo] for

a complete discussion. In this paper, we consider only fuzzy

rules in the fbrm of (1).

The fuz;$cation interjiace calculate the membership function of an input to the fuzzy sets of the system. Specifically,

j’B(dY d?/

yo = d e f u z z ( B ) =

j’w

dY

b ) Max-Criteria:

y, = d e f u z z ( B ) = y

such that B(y) is maximum.

c ) Mean of maximum [7]:

c

y

Yo

=

lw I

where W = {y/B(y) is maximum}.

More in general, we can define a defuzzification method as

a mapping from the fuzzy subsets of V into V , V being the

universe of the output variable y.

Thus, a function can be associated to each Fuzzy Logic

Controller as follows:

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zyx

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CASTRO FUZZY LOGIC CONTROLLERS ARE UNIVERSAL APPROXIMATORS

631

a ) Fuu$cation: A fuzzy set A' = Fuzzy(x,) is associated to the input xo.

b) Fuuy Inference: An approximate output is obtained by

fuzzy reasoning:

will be applied if and only if the input 2 = B matches

with the antecedent, i.e. iff A j ( B ) # 0, being Aj(2) =

T(Alj(X1), A2j(22), ... &j(zn)).

b) If the input 2" matches with the antecedent, the inference

is

B' = Fuzzy Inference from A'

c ) Defuuijication: The output value is obtained from the

approximate output:

z1 is A1 and 2 2 is A2 and . . .xn is A,, then y is B

Zis A'

y is B'

B'(y) = sup {T'(A'(Z), I(A(Z), B ( y ) ) ) / ZE R"}.

A (2)= T(Al(zl),A2(22),...,An(x")),

(3)

zyx

zyxwvutsr

yo = D e f u z z ( B ' ) .

111. TYPES OF FLc SYSTEMS WHICH

ARE UNIVERSAL

APPROXIMATORS

The general question about approximation is the following:

Let consider that a type of FLC,i.e. a fuzzification method,

a fuzzy inference method, a defuzzification method, and a class

of fuzzy rules RUL, are fixed. Given an arbitrary continuous

real valued function f on a compact U c R", and a certain

E > O , is it possible to find a set of fuzzy rules in RUL such

that the associated fuzzy controller approximates f to level E?

Specifically, we have looked for types of FLCs such that the

answer to the above question is positive. The main result we

present here is that the approximation is possible for almost

any type of fuzzy logic controller. We will carry out the proof

of this result in two cases: i) FLCs with fuzzy consequent and

ii) FLC's with non fuzzy consequent.

For each a < b E R let p(a, b) : R --+ R be a membership

function such that p(a, b)(x) # 0 iff z E (a, b). Let T and T'

be two t-norms, I a fuzzy implication, and T* a t-conorm.

and as the input is a point 2 = B

{

A'(2) = 1 if 2 = 1

0 in other case

the result will be is translated into

c) In general, we can express the inference of the rule Rj

when the input is 3c' = P by:

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A. FLCs with Fuzzy Consequent are Universal Approximators

Let SI = &(TI, T , I, T*, p ( a , b ) ) be the family of all

FLC systems where:

i) The fuzzification method is the point fuzzification

ii) The rules base is composed of a finite number of rules

with the form

If z1 is A1 and . . . x, is A,, then y is B.

where the membership functions of each Aij is ~ ( u ;a;j)

~ ,for

some aij 0 there exists a S, E S1 such that

Lemma I : Under the conditions of theorem 1 there exists

a S, E S1 such that

B'(y)lf(?) - yI

5 B'(y) * E, for eachy E R.

632

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IEEE TRANSACTIONS ON SYSTEMS. MAN, AND CYBERNETICS, VOL 25. NO 4, APRIL 1995

Proof: Let a' E U be. As f is continuous at a', for each

. . . n there exists a :6 > 0 such that

i = 1

and A; = Xl/C, X j .

In this case the general expression of B' will be

For each a' E U , set

Then, 0,-is an open on R" and U C U z E OZ,

~ since a' E 0,for each a' E Lr. As U is compact, there exist a finite subfamily

0~1,

0 5 2 , . . ., 0,-L

such that

U

c 0a'l U

iv') The defuzzification is 6 = C Xiw;, where

Oa'2

U .. .UO$.

B;(Y)

=

if Aj(.") = 0 or y # w

I ( A j ( P )w

. j ) in other case

The only parameters not specified in this class are the

number of rules k , such that j = 1, . . ., k ; and those describing

the membership functions,

and w, ( i = 1 . . . 7 1 , j 1

. ..k ) .

Set S, E S1 defined by

Theorem 2 ) Let f : U C R" .+ R be a continuous function

defined on a compact U . For each t > 0 there exists a S , E S 2

such that

sup {If(.' )

and

- Se(.')1/. E U }

I 6.

Lemma 2 ) Under the conditions of theorem 2 there exists

a S, E S 2 such that

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Now:

If B'(y) = 0, the lemma is trivial.

If B'(y) > 0 , then B'(y) = T*((B:(y))3,1

k ) > O . hence

there exists a j , such that B,(y) > 0. We have that:

a) From 13,(y) # 0, it follows A J o ( Z 0 )# 0, and as

AJO(Zo)= T ( A i , ~ ( . ~ ) , A 2 3 ~ ( . 0 2 ) . . . , A n, lit~follows

(z~))

for each y E R.

Proof: Let a' E U be, As f is continuous at a', for each

i = 1 . . . n there exists a 6; > 0 such that

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b) From I3,(y) # 0, it follows B,(y) = I(AJ0(."), For each a' E U , set

B J 0 ( y ) )# 0. If B,.,(Y) = 0, then B:o(y) = I ( A J O ( Z o ) ,

BJ0(y)) = I (.4,,,(."), 0) = 0, (from the property of the chosen

implication). Thus, from B, (y) # 0, it follows BJ0(y) # 0.

Then, 0~is an open on R" and U C U Z ~ ~since

O ~Fl ,E 0,From b), it follows I f ( + ) - yI I t / 2 .

From a), it follows Izp - $1 < Si,, ( i = 1 .. . n ) and for each a' E U . As U is compact, there exist a finite subfamily

0,-1,OZ2, . . ., OZ' such that

If(.")

- f(a''0)I

I 42.

Hence

Let S, E

Q.E.D.

Proof of thr Theorem 1: According to our hypotheses,

S,(.") belongs to the support of B' as S , ( P ) = d e f u z z ( B ' ) .

Hence B'(S,(.'O)) > 0 , and thus from B'(y)l(f(z,)

- y)I 5

B'(y) * t, we can conclude l(f(.")

- S,(Zo)

5Jt

Q.E.D.

B. FLCs with Non-Fuzzy Consequent are

Universal Approximators

Let S 2 = &(T', T , I , p ( a , b ) ) defined as S I , but exchanging ii) with ii') and iv) by id):

ii') The rules base is composed by a finite number of rules

with the form

Ifzl isA1 andx2isAzand ... z,isA,.

defined by

and

Wj(Y)

= f ( 3 )( j = 1 . , . k ) .

Now:

If B'(y) = 0, the lemma is trivial.

If B'(y) > 0. then B'(y) = T*((B:(y)),,lk ) > 0. hence

there exists a j , such that B,(y) > 0. We have that:

a) From B,(y) # 0, it follows AJ0(.'O) # 0.

b) From B,(y) # 0, it follows y = wJ0= f ( S i 3 0 ) . From a),

it follows l z ~ - u j " l< 5,kJ0(z = 1 . . . n ) and If(.'o)-f(a'Jo)I I

t. Hence

zyxwvuts

thenyisw:

where the membership functions of each A,, is p(&,

some a!.G J

zyxwvutsrq

629

IEEE TRANSACTIONS ON SYSTEMS,MAN,AND CYBERNETICS, VOL 25, NO. 4, APRIL 1995

Fuzzy Logic Controllers Are

Universal Approximators

zyxwvutsrqpo

zyxw

zyxwvuts

J. L. Castro

Abstract-In this paper, we consider a fundamental theoretical

question, Why does fuzzy control have such good performance

for a wide variety of practical problems?. We try to answer this

fundamental question by proving that for each fixed fuzzy logic

belonging to a wide class of fuzzy logics, and for each fixed type

of membership function belonging to a wide class of membership

functions, the fuzzy logic control systems using these two and any

method of d e f d c a t i o n are capable of approximating any real

continuous function on a compact set to arbitrary accuracy. On

the other hand, this result can be viewed as an existence theorem

of an optimal fuzzy logic control system for a wide variety of

problems.

1

I. INTRODUCTION

D

Fig. 1. Triangular membership function.

Manuscript received August 14, 1992; revised August 1, 1993 and June

3, 1994.

J. L. Castro is with the Department of Computer Science and Artificial

Intelligence, Universidad de Granada, 18071 Granada, Spain.

IEEE Log Number 9406630.

i.e., they are capable of approximating any real continuous

function on a compact set to arbitrary accuracy. This class is

that with:

1) Gaussian membership functions,

2) Product fuzzy conjunction,

3) Product fuzzy implication,

4) Center of area defuzzification.

Other approaches are due to Buckley [4], [5]. He has

proved that a modification of Sugeno type fuzzy controllers

are universal approximators. The modifications are:

1) The consequent part of the rules are polynomial functions, not only linear functions as in Sugeno type controllers,

2) The defuzzification is 6 = CXip(ni,m;),where X i is

the matching of the input value with the antecedent

part of the rule Ri, while in the Sugeno controller it

is X i = X;/CXj.

Although both results are very important, many real fuzzy

logic controllers do not belong to these classes. The main

reasons are that other membership functions are used, other

inference mechanisms are applied or other type of rules are

used.

The most common membership functions are the triangular

(see Fig. 1) or trapezoidal (see Fig. 2) functions. With respect

the fuzzy inference, a wide variety of fuzzy implications are

used: R-implications [21] and Mamdani implication [ 121 are

the most common. Finally, in many fuzzy controllers the

consequent part is not a polynomial function but a fuzzy

proposition or a linear or constant function.

URING the past several years, fuzzy logic control (FLC)

has been successfully applied to a wide variety of practical problems. Notable applications of that FLC systems include

the control of warm water [7], robot [6], heat exchange [15],

traffic junction [16], cement kiln [9], automobile speed [14],

automotive engineering [25], model car parking and turning

[19], [20], turning [17], power system and nuclear reactor [3],

etc . ..

It points out that fuzzy control has been effectively used in

the context of complex ill-defined processes, specially those

which can be controlled by a skilled human operator without

the knowledge of their underlying dynamics. In this sense, neural and adaptive fuzzy systems has been compared to classical

control methods by B. Kosko in [8]. There, it is remarked that

they are model-free estimators, i.e., they estimate a function

without requiring a mathematical description of how the output

functionally depends on the input; they learn from samples.

However, some people criticize fuzzy control because its

effectiveness has not been proved. That is, the very fundamental theoretical question “Why does a fuzzy rule-based system

have such good performance for a wide variety of practical

problems?’ remains unanswered. There exist some qualitative

explanations, e.g., “fuzzy rules utilize linguistic information”,

“fuzzy inference simulates human thinking procedure”, “fuzzy

rule systems capture the approximate and inexact nature of

the real world,” etc., but mathematical proofs have not been

obtained.

A first approach to answer this fundamental question in a

quantitative way was presented by Wang [ 181. He proved that

a particular class of FLC systems are universal approximators,

a

b

C

zyxw

zyxw

zyxwvutsr

0018-9472/95$04.00 0 1995 IEEE

630

zyxwvutsrqponmlkj

zyxwvu

zyxwv

zyx

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 25, NO. 4, APRIL 1995

if U is the universe of an input variable X, and X = 5 , E U

is the input value, the output of the fuzzification interface is a

fuzzy set on U , F = Fuzzy(z,). There are two main types

of fuzzification:

zyx

A) Point defuzzijication:

zyx

1 if z = x,

0 in other case

a

Fig. 2.

b

C

d

B ) Approximate defuuijkation:

Trapell iidal membenhip function.

Thus, the question: “Are these other types of fuzzy logic

controllers (which are usually applied) also universal approximators?” or, In a more general form “What other types of fuzzy

logic controllers are universal approximators?’ still remains

unanswered.

In this papcr we will answer this question for a large number

of cases. Specifically, we will prove that other classes of FLC

are also universal approximators. These classes are those with:

1) A kind of membership functions, including among others, trapezoidal or triangular membership functions,

2 ) the fuzLy conjunction modeled by an arbitrary t-norm,

3) the fuzzy implication only needs to satisfy a weak

property (R-implications and t-norms satisfy it),

4) the defuzzification method only needs to satisfy a very

weak condition (usual defuzzification methods satisfy it).

Many of examples cited in the first paragraph belong to one

of these clases. Yamakawa’s [23], Expert Systems [25],and

Sugeno’s [ 191, [20] fuzzy controllers belong to these classes.

( z - x,\ < S.

F ( z ) # 0 if and only if

There are many different kinds of fuzzy logic which may

be used in a fuzzy inference machine. The general inference

rule from a single rule

if X is A then Y is B

(2)

is the generalized modus ponens [ 131:

if X is A

X is A’

Y is B

then

Y is B‘

where

zyxw

zyxwvut

B’(y) = supT’(A’(~).I ( A ( . E )B. ( y ) ) )

X

depending on a t-norm T’ and an implication function 1.To

translate a fuzzy rule of kind (1) to a simple fuzzy rule of type

( 2 ) is used a fuzzy conjunction:

X Iis A1 and X2 is A2 and . . . X,,is A,. then y is B

ZisA

where A(.?) = T(A1(xl). A ~ ( z * ) . . . . A,(z,)) depending on

a t-norm T.

The defuzziJication interface defuzzies the fuzzy output of

a system to generate a non fuzzy output. The most commonly

used defuzzification methods are:

a ) Center of Area:

zyxwvutsrqp

zyxwvutsrq

zyxwvutsrqpo

11. FUZZYLOGICCONTROLLERS

A FLC system is composed by four principal elements: fuzzy

rule base, fiuzification interface, fuzzy inference machine,

and defuzzification interface. In this paper we consider multiinput-single-output (MISO) fuzzy logic control systems f :

U C 72” + ’R, because a multi-output system can always be

separated into a collection of single-output systems.

The fuzzy rule base is a set of linguistic statements in

the form: “IF a set of conditions are satisfied THEN a set

of consequences are inferred”, where the conditions and the

consequences are associated with fuzzy concepts (i.e. linguistic

terms). For example, in the case of a MISO FLC with n inputs,

the fuzzy rule base may consist of the following rules:

Rj:

If:rlisAiarid . . . aridx,isAA. t h e n y i s B J

(1)

( i =: 1 . . . U ) are the inputs to the fuzzy rule system,

where

y is the output of the system, A i and BJ (j= 1 . . . k )

are the linguistic terms, and k is the number of fuzzy rules.

By relating each linguistic term in the fuzzy rules with a

membership function, we specify the meaning of the rules.

There are many different kinds of fuzzy rules: see [lo] for

a complete discussion. In this paper, we consider only fuzzy

rules in the fbrm of (1).

The fuz;$cation interjiace calculate the membership function of an input to the fuzzy sets of the system. Specifically,

j’B(dY d?/

yo = d e f u z z ( B ) =

j’w

dY

b ) Max-Criteria:

y, = d e f u z z ( B ) = y

such that B(y) is maximum.

c ) Mean of maximum [7]:

c

y

Yo

=

lw I

where W = {y/B(y) is maximum}.

More in general, we can define a defuzzification method as

a mapping from the fuzzy subsets of V into V , V being the

universe of the output variable y.

Thus, a function can be associated to each Fuzzy Logic

Controller as follows:

zyxwvutsrqponml

zyx

zyxwvutsrqpo

zyxwvutsrq

zyxwvuts

CASTRO FUZZY LOGIC CONTROLLERS ARE UNIVERSAL APPROXIMATORS

631

a ) Fuu$cation: A fuzzy set A' = Fuzzy(x,) is associated to the input xo.

b) Fuuy Inference: An approximate output is obtained by

fuzzy reasoning:

will be applied if and only if the input 2 = B matches

with the antecedent, i.e. iff A j ( B ) # 0, being Aj(2) =

T(Alj(X1), A2j(22), ... &j(zn)).

b) If the input 2" matches with the antecedent, the inference

is

B' = Fuzzy Inference from A'

c ) Defuuijication: The output value is obtained from the

approximate output:

z1 is A1 and 2 2 is A2 and . . .xn is A,, then y is B

Zis A'

y is B'

B'(y) = sup {T'(A'(Z), I(A(Z), B ( y ) ) ) / ZE R"}.

A (2)= T(Al(zl),A2(22),...,An(x")),

(3)

zyx

zyxwvutsr

yo = D e f u z z ( B ' ) .

111. TYPES OF FLc SYSTEMS WHICH

ARE UNIVERSAL

APPROXIMATORS

The general question about approximation is the following:

Let consider that a type of FLC,i.e. a fuzzification method,

a fuzzy inference method, a defuzzification method, and a class

of fuzzy rules RUL, are fixed. Given an arbitrary continuous

real valued function f on a compact U c R", and a certain

E > O , is it possible to find a set of fuzzy rules in RUL such

that the associated fuzzy controller approximates f to level E?

Specifically, we have looked for types of FLCs such that the

answer to the above question is positive. The main result we

present here is that the approximation is possible for almost

any type of fuzzy logic controller. We will carry out the proof

of this result in two cases: i) FLCs with fuzzy consequent and

ii) FLC's with non fuzzy consequent.

For each a < b E R let p(a, b) : R --+ R be a membership

function such that p(a, b)(x) # 0 iff z E (a, b). Let T and T'

be two t-norms, I a fuzzy implication, and T* a t-conorm.

and as the input is a point 2 = B

{

A'(2) = 1 if 2 = 1

0 in other case

the result will be is translated into

c) In general, we can express the inference of the rule Rj

when the input is 3c' = P by:

zyxwvutsr

zyxwvutsrq

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A. FLCs with Fuzzy Consequent are Universal Approximators

Let SI = &(TI, T , I, T*, p ( a , b ) ) be the family of all

FLC systems where:

i) The fuzzification method is the point fuzzification

ii) The rules base is composed of a finite number of rules

with the form

If z1 is A1 and . . . x, is A,, then y is B.

where the membership functions of each Aij is ~ ( u ;a;j)

~ ,for

some aij 0 there exists a S, E S1 such that

Lemma I : Under the conditions of theorem 1 there exists

a S, E S1 such that

B'(y)lf(?) - yI

5 B'(y) * E, for eachy E R.

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IEEE TRANSACTIONS ON SYSTEMS. MAN, AND CYBERNETICS, VOL 25. NO 4, APRIL 1995

Proof: Let a' E U be. As f is continuous at a', for each

. . . n there exists a :6 > 0 such that

i = 1

and A; = Xl/C, X j .

In this case the general expression of B' will be

For each a' E U , set

Then, 0,-is an open on R" and U C U z E OZ,

~ since a' E 0,for each a' E Lr. As U is compact, there exist a finite subfamily

0~1,

0 5 2 , . . ., 0,-L

such that

U

c 0a'l U

iv') The defuzzification is 6 = C Xiw;, where

Oa'2

U .. .UO$.

B;(Y)

=

if Aj(.") = 0 or y # w

I ( A j ( P )w

. j ) in other case

The only parameters not specified in this class are the

number of rules k , such that j = 1, . . ., k ; and those describing

the membership functions,

and w, ( i = 1 . . . 7 1 , j 1

. ..k ) .

Set S, E S1 defined by

Theorem 2 ) Let f : U C R" .+ R be a continuous function

defined on a compact U . For each t > 0 there exists a S , E S 2

such that

sup {If(.' )

and

- Se(.')1/. E U }

I 6.

Lemma 2 ) Under the conditions of theorem 2 there exists

a S, E S 2 such that

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Now:

If B'(y) = 0, the lemma is trivial.

If B'(y) > 0 , then B'(y) = T*((B:(y))3,1

k ) > O . hence

there exists a j , such that B,(y) > 0. We have that:

a) From 13,(y) # 0, it follows A J o ( Z 0 )# 0, and as

AJO(Zo)= T ( A i , ~ ( . ~ ) , A 2 3 ~ ( . 0 2 ) . . . , A n, lit~follows

(z~))

for each y E R.

Proof: Let a' E U be, As f is continuous at a', for each

i = 1 . . . n there exists a 6; > 0 such that

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b) From I3,(y) # 0, it follows B,(y) = I(AJ0(."), For each a' E U , set

B J 0 ( y ) )# 0. If B,.,(Y) = 0, then B:o(y) = I ( A J O ( Z o ) ,

BJ0(y)) = I (.4,,,(."), 0) = 0, (from the property of the chosen

implication). Thus, from B, (y) # 0, it follows BJ0(y) # 0.

Then, 0~is an open on R" and U C U Z ~ ~since

O ~Fl ,E 0,From b), it follows I f ( + ) - yI I t / 2 .

From a), it follows Izp - $1 < Si,, ( i = 1 .. . n ) and for each a' E U . As U is compact, there exist a finite subfamily

0,-1,OZ2, . . ., OZ' such that

If(.")

- f(a''0)I

I 42.

Hence

Let S, E

Q.E.D.

Proof of thr Theorem 1: According to our hypotheses,

S,(.") belongs to the support of B' as S , ( P ) = d e f u z z ( B ' ) .

Hence B'(S,(.'O)) > 0 , and thus from B'(y)l(f(z,)

- y)I 5

B'(y) * t, we can conclude l(f(.")

- S,(Zo)

5Jt

Q.E.D.

B. FLCs with Non-Fuzzy Consequent are

Universal Approximators

Let S 2 = &(T', T , I , p ( a , b ) ) defined as S I , but exchanging ii) with ii') and iv) by id):

ii') The rules base is composed by a finite number of rules

with the form

Ifzl isA1 andx2isAzand ... z,isA,.

defined by

and

Wj(Y)

= f ( 3 )( j = 1 . , . k ) .

Now:

If B'(y) = 0, the lemma is trivial.

If B'(y) > 0. then B'(y) = T*((B:(y)),,lk ) > 0. hence

there exists a j , such that B,(y) > 0. We have that:

a) From B,(y) # 0, it follows AJ0(.'O) # 0.

b) From B,(y) # 0, it follows y = wJ0= f ( S i 3 0 ) . From a),

it follows l z ~ - u j " l< 5,kJ0(z = 1 . . . n ) and If(.'o)-f(a'Jo)I I

t. Hence

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where the membership functions of each A,, is p(&,

some a!.G J