Fuzzy Logic Controllers Are Universal Approximators

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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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0018-9472/95$04.00 0 1995 IEEE

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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:

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A) Point defuzzijication:

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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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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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CASTRO FUZZY LOGIC CONTROLLERS ARE UNIVERSAL APPROXIMATORS

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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)

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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.

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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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thenyisw:

where the membership functions of each A,, is p(&,
some a!.G J