IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 8, NO. 6, DECEMBER 2000
773
A Comparative Study on Sufficient Conditions for Takagi–Sugeno Fuzzy
Systems as Universal Approximators
Ke Zeng, Nai-Yao Zhang, and Wen-Li Xu
Abstract—Universal approximation is the basis of theoretical research and practical applications of fuzzy systems. Studies on the
universal approximation capability of fuzzy systems have achieved
great progress in recent years. In this paper, linear Takagi–Sugeno
(T–S) fuzzy systems that use linear functions of input variables as
rule consequent and their special case named simplified fuzzy systems that use fuzzy singletons as rule consequent are investigated.
On condition that overlapped fuzzy sets are employed, new sufficient conditions for simplified fuzzy systems and linear T–S fuzzy
systems as universal approximators are given, respectively. Then,
a comparative study on existing sufficient conditions is carried out
with numeric examples.
Index Terms—Linear Takagi–Sugeno (T–S) fuzzy systems, simplified fuzzy systems, sufficient condition, universal approximation.
T
HE question of universal approximation means determining whether fuzzy systems can uniformly approximate
any real continuous function on a compact set to arbitrary
degree of accuracy. From a mathematical point of view, fuzzy
systems perform a mapping from input space to output space.
Universal approximation capability of fuzzy systems is the
basis of almost all the theoretical research and practical applications of fuzzy systems, e.g., fuzzy control, fuzzy identification,
fuzzy expert system, and so on. Hence, studying universal
approximation capability of fuzzy systems is very necessary
and important both in theory and in practice.
Studies on universal approximation of fuzzy systems have
achieved great progress in the past few years. Many different
fuzzy systems have been discussed. However, in reality, general fuzzy systems are mainly of two classes: Mamdani fuzzy
systems and Takagi–Sugeno (T–S) fuzzy systems [1]. The main
difference between these two classes lies in their consequent of
the fuzzy rules. Mamdani fuzzy systems use fuzzy sets as rule
consequent. The th rule of the Mamdani fuzzy systems can be
expressed as
is
and
and
is
where
input variable
input fuzzy set;
IF
is
and
is
and
and
is
THEN
where
is a linear or nonlinear function of input variables.
Generally, linear function of input variables is chosen, i.e.,
IF
is
and
is
and
and
is
THEN
I. INTRODUCTION
IF
is
and
THEN is
output variable;
output fuzzy set;
number of fuzzy rules.
However, T–S fuzzy systems use functions of input variables
as rule consequent. The th rule of T–S fuzzy systems can be
written as
;
Manuscript received February 28, 2000; revised July 28, 2000. This work was
supported by the National Natural Science Foundation of China under Grant
69774015 and the Research Foundation of Information School, Tsinghua University, Beijing, China.
The authors are with the Department of Automation, Tsinghua University,
Beijing 100084, China (e-mail: [email protected]).
Publisher Item Identifier S 1063-6706(00)10689-7.
(1)
. We call T–S fuzzy systems defined by (1) linear
where
T–S fuzzy systems. Although any linear or nonlinear function
, linear T–S fuzzy systems are commonly
can be chosen as
used in research and applications, because choosing nonlinear
functions of input variables may cause a lot of troubles in analysis and design of fuzzy system.
An interesting fact we would like to point out here is that
Mamdani fuzzy systems employing fuzzy singletons as rule
consequent are equivalent to linear T–S fuzzy systems with
. Therefore, we call this type of
Mamdani fuzzy systems simplified fuzzy systems. The th rule
of simplified fuzzy systems is as follows:
IF
is
THEN
and
is
and
and
is
(2)
In the past few years, many authors have addressed the
question of universal approximation by fuzzy systems [2]–[19].
These authors discussed different types of fuzzy systems with
different restrictions. Yet, most of the existing results in literatures on fuzzy systems as universal approximators deal with
Mamdani fuzzy systems [2], [3], [7], [8], [10]–[13], [15]–[19].
For example, Wang [13] proved that Mamdani fuzzy systems
with Gaussian membership functions, product inference, and
centroid defuzzification are universal approximators. Under
otherwise equal conditions, Zeng [18], [19] proved that Mamdani fuzzy systems with pseudotrapezoid-shaped membership
functions are universal approximators. Note that almost all
the Mamdani fuzzy systems discussed in above references
use fuzzy singletons as rule consequent, so, actually they are
simplified fuzzy systems.
Comparing with Mamdani fuzzy systems, T–S fuzzy systems
are more widely used in control and identification, whereas up
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 8, NO. 6, DECEMBER 2000
until now only a few results dealing with T–S fuzzy systems
as universal approximators can be found in [4]–[6], [9], and
[14]. For example, two-input single-output T–S fuzzy systems
with linear defuzzifier (weighted sum) instead of the commonly
used centroid defuzzifier (weighted average) are proved to be
universal approximators in [9]. T–S fuzzy systems using proportional linear functions of input variables as rule consequent
are proved to be universal approximators in [4], [5]. Linear T–S
fuzzy systems with full-overlapped membership functions have
been proved to be universal approximators in [6].
So far, the majority of existing literatures concentrated on answering the “existence problem”; that is, given any real continuous function on a compact set, there exists certain fuzzy system
that can uniformly approximate the function to any degree of
accuracy. Nevertheless, we are far from being satisfied. An important question followed is “given a real continuous function,
how to design the membership functions and how many fuzzy
sets are needed for each input variable, in order to guarantee
the desired approximation accuracy?” The answer to this question is called sufficient condition for fuzzy systems as universal
approximators. Only a few papers [3]–[6], [15], [20] explicitly
dealt with sufficient conditions, where [4] and [5] are on linear
T–S fuzzy systems using proportional linear functions of input
variables as rule consequent, [6] deals with linear T–S fuzzy
systems with full-overlapped membership functions, and [3],
[15], and [20] are on simplified fuzzy systems using pseudotrapezoid-shaped membership functions. We would like to point
out that among [3]–[6], [15], [20], the majority of Ying’s results
deals with fuzzy systems using general fuzzy logic operators
and general defuzzifier.
In this paper, two new sufficient conditions for simplified
fuzzy systems and linear T–S fuzzy systems as universal approximators are given, respectively, and a comparative study on
existing sufficient conditions is carried out with numeric examples. The structure of the rest of this paper is as follows. After
the introduction, some mathematical preliminaries are given in
Section II. Then, new sufficient conditions for simplified fuzzy
systems and linear T–S fuzzy systems as universal approximators are given in Section III and Section IV, respectively. The
comparative study on different sufficient conditions is carried
out in Section V. We conclude this paper with Section VI.
II. MATHEMATICAL PRELIMINARIES
In the rest of this paper, we always assume that infinite norm
is defined as
, where the
.
compact set
A multivariate polynomial of degree defined on compact set
can be expressed as [6]
where
.
1) Weierstrass Approximation Theorem [21]: There always
on a compact set
, which
exists polynomial
can uniformly approximate any given real continuous function
Fig. 1. Illustration of the choice of central point of a fuzzy set A.
Fig. 2. A group of overlapped fuzzy sets A (i = 1; 2; . . . ; n).
on
to arbitrary accuracy, i.e.,
.
Markov Theorem [21]: For a polynomial
, suppose that
, then
such that
defined on
Definition 1—Support [19]: Denote the support of a fuzzy
to be
.
set on
Definition 2—Core [19]: Denote the core of a fuzzy set
on
to be
, which is a subset
.
of support
is said
Definition 3—Normal [19]: A fuzzy set on
.
to be normal if
on
Definition 4—Central Point: For a normal fuzzy set
, one and only one
should be chosen as the
central point of fuzzy set .
only
Remark 1: For the most popular case, i.e.,
(see Fig. 1), then
and
for
. Generally
is chosen to be the central
point of fuzzy set . However, other choices are acceptable, too,
, etc.
such as
Definition 5—Overlapped: A group of normal fuzzy sets
defined on
is said to be overlapped
hold for any
if
, where
is the central point of fuzzy set
.
Fig. 2 shows a group of overlapped fuzzy sets.
Remark 2: Recall the definition of “full-overlapped” it reand
hold for
quires
. Apparently, the definition of “overlapped”
any
is a natural expansion of “full-overlapped” and is more flexible
in the design of fuzzy system.
defined on
Remark 3: If a group of fuzzy sets
is overlapped (see Fig. 2), we can easily verified that
hold and, for
for any
, at most two adjacent fuzzy sets
and
can
any
have memberships of greater than zero and there must have
at the same time
.
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 8, NO. 6, DECEMBER 2000
775
where
is the
. And the output of
activation degree of fuzzy rule
simplified fuzzy systems can be expressed as
(4)
Fig. 3. Illustration of a group of equally distributed fuzzy sets of x .
Without loss of generality, we assume that
(because a simple linear transform can map
to
). Further, we assume the central point of the th
is chosen to locate at
fuzzy set of
and these central points have the arrangement that
. For each input variable
, we define distance of fuzzy partition
where
. Based on this definition, we define
maximum distance of fuzzy partition
Definition 6—Equally Distributed: A group of fuzzy sets asis said to be equally distributed if
signed to input variable
are equal to each other.
Remark 4: It can be easily verified that no matter how
is partitioned,
the universe of discourse of input variable
always holds and
holds if and only if equally distributed fuzzy sets are employed
(see Fig. 3).
of linear T–S fuzzy systems given by
Observing fuzzy rule
, we can find out that this rule determines
(1)
, where
a special input vector
is exactly equal to the corresponding central point
, i.e.,
. It is obvious that
of
special input vectors in all and these vectors are
there are
different from each other according to different fuzzy rules. We
define the collection of these special input vectors as
III. SUFFICIENT CONDITION FOR SIMPLIFIED FUZZY SYSTEMS
AS UNIVERSAL APPROXIMATORS
In this section, we will prove a new sufficient condition for
simplified fuzzy systems as universal approximators in order
to answer the question “given a real continuous function, how
to design the membership functions and how many fuzzy sets
are needed for each input variable hence, to ensure the desired
approximation accuracy?”
overlapped and equally disLemma 1: Suppose that
tributed fuzzy sets are assigned to each input variable of
simplified fuzzy systems, then for any given polynomial
defined on
and approximation error bound
, there exists a simplified fuzzy system such that
holds when
Proof: Observing each fuzzy rule
fired by , i.e.,
, this fuzzy rule determines a speof simplified
cial input vector
fuzzy systems. Since the fuzzy sets assigned to each input variable are overlapped and equally distributed for any component
of , according to Remark 3 and Remark 4,
we have
On the other hand, also from Remark 3, it is easy to verify
, only fuzzy rule
that, for any
can be fired, and
hold for any
. Hence, from (4) we have
Let
where
.
Since the difference between simplified fuzzy systems and
linear T–S fuzzy systems only lies in their rule consequent, the
same set of special input vectors can also be defined for simplified fuzzy systems.
fuzzy sets are assigned to input variable ,
Suppose that
. Using
therefore, the number of fuzzy rules is
product inference and centroid defuzzifier, the output expression of linear T–S fuzzy systems can be written as
Note that is only concerned with
and and is irrelevant
with , hence, it is constant.
Utilizing Taylor’s formula with Lagrange remainder, we have
where the Lagrange remainder
can be expressed as
(3)
and locates in the minimum super sphere containing
and
.
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IV. SUFFICIENT CONDITION FOR LINEAR T–S FUZZY SYSTEMS
AS UNIVERSAL APPROXIMATORS
Thus
Therefore, for any input vector
simplified fuzzy systems, from (4) we have
of
Since simplified fuzzy systems are special case of linear T–S
fuzzy systems, obviously, as a sufficient condition, Lemma 1
and Theorem 1 is also applicable for linear T–S fuzzy systems,
will hold when
e.g.,
. Unfortunately, it is really “excessively sufficient.” For example, given a polynomial
defined on
, we use two-input single-output linear T–S
fuzzy systems to approximate it. According to this result, we
. However, suppose that each rule consequent
have
, thereof the linear T–S fuzzy system is chosen as
fore, it can be easily verified that
Obviously, any positive integer
is acceptable. As it
is well known that generally T–S fuzzy systems are more capable of approximating functions than simplified fuzzy systems,
this example provides a nice demonstration and can serve as
our starting point to find a more precise sufficient condition for
linear T–S fuzzy systems.
overlapped and equally disLemma 2: Suppose that
tributed fuzzy sets are assigned to each input variable of
linear T–S fuzzy systems, then for any given polynomial
defined on
and universal approximation error
, there exists a linear T–S fuzzy system such that
bound
holds when
Hence, when
will hold.
Q.E.D.
overlapped and equally disTheorem 1: Suppose that
tributed fuzzy sets are assigned to each input variable of simplified fuzzy systems, then for any given real continuous funcdefined on
and approximation error bound
tion
, a simplified fuzzy system exists which can guarantee
if
where
and
holds.
Proof: First, according to Weierstrass approximation thesuch that
orem, there always exists a polynomial
holds.
Second, from Lemma 1, a simplified fuzzy system exists,
if
which can guarantee
. This implies that
Proof: For any input vector
linear T–S systems, from (3) we have
of
Observing each fuzzy rule
fired
, this fuzzy rule determines a special input vector
of linear T–S fuzzy systems.
Since the fuzzy sets of each input variable are overlapped and
equally distributed, according to Remark 3 and Remark 4, for
of we have
any component
by
On the other hand, also from Remark 3, it is easy to verify
, only fuzzy rule
that for any
can be fired and
hold for any
. Hence, it can be easily verified that
Q.E.D.
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 8, NO. 6, DECEMBER 2000
Utilizing Taylor’s formula with Lagrange remainder, we have
where the Lagrange remainder
777
bound
antee
, a linear T–S fuzzy system exists which can guarif
can be expressed as
and locates in the minimum super sphere containing
is the Hessian matrix of
.
Let
and
where
. It is obvious that this equation has a unique
. Note
set of solutions
and , and are
that the solutions are only concerned with
irrelevant with input variable , hence, they are constants.
Now we have
where
and
holds.
The proof of Theorem 2 is omitted since it is similar to Theorem 1.
Remark 6: Theorems 1 and 2 mathematically explain, to
some extent, the well-known fact that, generally, T–S fuzzy
systems are more capable of approximating functions than
simplified fuzzy systems. Common explanation of this fact
is that T–S fuzzy system can use different hyperplanes to
approximate function whereas simplified fuzzy system can
and
use only horizontal hyperplanes. Since
are finite numbers, when
is chosen
is
for linear T–S fuzzy
small enough, we have
is
for
systems as universal approximators, whereas
simplified fuzzy systems. As seen, less fuzzy sets are needed
by linear T–S fuzzy systems comparing with simplified fuzzy
systems. This implies immediately smaller number of fuzzy
rules, smaller system structure, faster inference rate, and faster
learning rate, etc.
V. COMPARATIVE STUDIES ON THE SUFFICIENT CONDITIONS
A. Implicit Sufficient Condition
Thus, see the equation at the bottom of the page.
Hence, on condition that
will hold.
Q.E.D.
Remark 5: Again, we use two-input single-output linear T–S
defined
fuzzy systems to approximate the polynomial
. This time, from Lemma 2, it is easy to compute the
on
. Since
should be a positive integer, we get
result
at last. This example demonstrates the correctness of
Lemma 2 from one aspect.
overlapped and equally disTheorem 2: Suppose that
tributed fuzzy sets are assigned to each input variable of linear
T–S fuzzy systems, then for any given real continuous funcdefined on
and universal approximation error
tion
Up to now, in the literatures some sufficient conditions for
fuzzy systems as universal approximators are proposed directly,
while others are implicit in the proof of fuzzy systems being capable of approximating functions. For example, Wang et al. [15]
proved multi-input single-output (MISO) simplified fuzzy systems with full-overlapped triangular membership functions are
universal approximators by simply using the concept of “uniform continuity.” In their proof, a sufficient condition for twoinput single-output case is actually acquired, i.e., the following
proposition.
Proposition: For any given real continuous function
on
and error bound
exists such
and
in the input
that for any two points
will hold as long as
space
. Therefore, by partitioning
equal intereach input universe of discourse into
vals, there exists a simplified fuzzy system that can uniformly
approximate function with error bound .
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The origin of the proof of this proposition can be found in
[10]. Although this proof, simply on the basis of uniform continuity, is very smart, the proposition obtained explicitly exand , but implicitly expresses the relationship between
and through the concept
presses the relationship between
the uniform continuity. It is not an intuitive form, hence, it is
difficult to verify and apply.
TABLE I
RESULTS OF COMPUTATION FOR DIFFERENT CASES
B. Computable Results
2) Ying’s Theorem on Simplified Fuzzy Systems: Ying
[3]–[6] first introduced a two-step method in deducing sufficient conditions for fuzzy systems as universal approximators,
i.e., first-prove fuzzy systems can uniformly approximate
polynomials, second-prove fuzzy systems can uniformly
approximate real continuous functions utilizing Weierstrass
approximation theorem. Our foregoing proofs in this paper
are also on the basis of this two-step method. The sufficient
conditions given by Ying are the first computable results. For
simplified fuzzy systems, Ying ([3], Theorem 2.4) obtained that
when the number of the fuzzy partitions of each input variable
satisfies the following formula:
the number of the fuzzy partitions of each input variable satisfies
the following formula:
(5)
will hold for a given polynomial
on
defined
.
Note that
(6)
will hold for a given continuous function
defined on
, where
.
Again we are going to prove that
(7)
and inequality (5) is obtained for polynomial
parently, we will have
on
, ap-
by
replacing the coefficients of
with their absolute
letting all the input variables
, where
values and
. This implies immediately that
So we can say that for the same case, the numerical result obtained by using Lemma 1 will be no greater than using inequality
(5). But, although Lemma 1 can get smaller result and is concise
in formula form, the computational complexity of Lemma 1 is
larger than inequality (5), especially for the high-dimensional
case, because it utilizes the extremum of the partial derivates of
polynomial.
3) Ying’s Theorem on Linear T–S Fuzzy Systems: Ying ([6],
Theorem 2) also obtained a sufficient condition for linear T–S
fuzzy systems with full-overlapped membership functions. His
result was given in two-input single-output case, that is, when
First, it is obvious that
Second, let us observe the computation result of
with
.
and
are exchangeable in this formula, we will
Note that
classify the computation results of this formula with different
and in the following five cases (see Table I). It is trivial to
is bigger. This is enough
verify that in all five cases
to prove inequality (7).
Hence, given the same target function to be approximated and
the same proper polynomial using Theorem 2 can in no case calculate a numerical result greater than using inequality (6). Furthermore, Theorem 2 is on condition that overlapped membership functions are employed and is much more concise. However, the main difference between Theorem 2 and inequality (6)
is small enough using Theorem 2 the result is of
is that if
precision, whereas using inequality (6) the result is
.
still of
4) Numeric Example: To intuitively show the difference,
here we present a numeric example, which was originally presented by Ying in [6] to verify the performance of inequality (6).
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 8, NO. 6, DECEMBER 2000
Example: What are the upper bounds on the number of fuzzy
sets assigned to each input variable for linear T–S fuzzy systems to uniformly approximate a function
with approximation error less than 0.2 or 0.1, where
?
on the interval
Solution 1: The function
can be approximated by a third-order polynomial
with truncation error slightly less than 0.071. Hence,
can be approximated uniformly by
the following third-order polynomial:
with truncation error slightly less than 0.071. Utilizing this polynomial to compute the minimal upper bounds on the number of
. For approximation acfuzzy rules, we will have
, [6] reported that 309 fuzzy sets are needed,
curacy
, it will increase to
whereas for approximation accuracy
1369.
Next, we calculate the result of Theorem 2. It is easy to get
that
779
dimensional case, the computational complexity of our Theorem 2 is much larger than Ying’s because it needs to compute the extremum of second-order partial derivatives whereas
inequality (6) only needs to calculate the coefficients of a polynomial.
C. Two Corollaries to Reduce the Computational Complexity
Inequality (7) shed some light on how to reduce the computational complexity of Theorem 2. That is, we can easily estimate
the upper bound of
for a polynomial
defined on
by simply calculates its coefficients, which
leads to the following corollary. And for sake of simplicity, this
corollary is given in the case of two input variables.
overlapped and equally disCorollary 1: Suppose that
tributed fuzzy sets are assigned to each input variable of linear
T–S fuzzy systems, then for any given real continuous function
defined on
and approximation error bound
,a
linear T–S fuzzy system exists, which can guarantee
if
where
and
holds.
Since our goal is to obtain sufficient condition, Corollary 1
can be acquired on the basis of Theorem 2 as long as we can
prove
Therefore, from Theorem 2, we have
For approximation accuracy
, our result is
,
hence, only six fuzzy sets are needed; for approximation accu, it is
, hence, only 14 fuzzy sets are
racy
needed.
on the interval
Solution 2: The function
can also be approximated by a fourth-order polynomial
with truncation error slightly less than 0.024. Hence utilizing
. This time, [6] reported
this polynomial, we have
while
that 351 fuzzy sets for approximation accuracy
. However,
809 fuzzy sets for approximation accuracy
from Theorem 2, only 8 and 11 fuzzy sets are needed, respectively.
As we have seen, our results are far smaller than Ying’s. But
the computational complexity problem still exists. In the high-
Take notice of Table I, this is trivial.
As Corollary 1 only needs to calculate the coefficients of a
polynomial, the computational complexity is significantly reduced comparing with Theorem 2. However, the defect is also
obvious; that is, Corollary 1 is not as precise as Theorem 2.
Especially, for the single-input single-output (SISO) case, we
utilizing Markov Theorem,
can get a better estimation of
which leads to Corollary 2.
overlapped and equally disCorollary 2: Suppose that
tributed fuzzy sets are assigned to the input variable of linear
T–S fuzzy systems, then for any given real continuous function
defined on
and approximation error bound
,
a SISO linear T–S fuzzy system exists which can guarantee
if
where
and
holds.
Since our goal is to obtain sufficient condition, Corollary 2
can be acquired on the basis of Theorem 2 so long as we can
prove
According to Markov theorem, this is sure.
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VI. CONCLUSION
In this paper, two sufficient conditions for simplified fuzzy
systems and linear T–S fuzzy systems to uniformly approximate
any real continuous function on a compact set to arbitrary degree of accuracy are deduced, respectively. Further, a comparative study on existing sufficient conditions is carried out with
numeric examples, which leads to two corollaries that can reduce the computational complexity of our theorem.
The following two problems are considered very important in
farther investigation:
1) to study the minimal upper bound on the number of fuzzy
sets assigned to each input variable for fuzzy systems as
universal approximators;
2) to study the lower bound on the number of fuzzy sets assigned to each input variable for fuzzy systems as universal approximators.
The answer to the latter problem is called necessary condition. Unfortunately, only Ying [22], [23] has addressed this
problem. His interesting works show that the minimal configurations of fuzzy systems depend on the number and locations of
the extremas of the function to be approximated.
Practically, to obtain the “necessary and sufficient condition”
for fuzzy systems as universal approximators is the final goal.
Since no necessary and sufficient condition has been acquired
for any type of fuzzy systems, we hope that by studying these
two problems, this goal will be eventually achieved.
REFERENCES
[1] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans. Syst., Man, Cybern.,
vol. SMC-15, pp. 116–132, 1985.
[2] B. Kosko, “Fuzzy system as universal approximators,” in Proc. IEEE
Int. Conf. Fuzzy Syst., San Diego, CA, Mar. 1992, pp. 1153–1162.
[3] H. Ying, “Sufficient conditions on general fuzzy systems as function
approximators,” Automatica, vol. 30, pp. 521–525, 1994.
, “General SISO Takagi-Sugeno fuzzy systems with linear rule con[4]
sequent are universal approximators,” IEEE Trans. Fuzzy Syst., vol. 6,
pp. 582–587, Nov. 1998.
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
, “General Takagi-Sugeno fuzzy systems with simplified linear rule
consequent are universal controllers, models and filters,” J. Inform. Sci.,
vol. 108, pp. 91–107, 1998.
, “Sufficient conditions on uniform approximation of multivariate
functions by general Takagi-Sugeno fuzzy systems with linear rule consequent,” IEEE Trans. Syst., Man, Cybern., vol. 28, pp. 515–520, 1998.
H. T. Nguyen, V. Kreinovich, and O. Sirisaengtaksin, “Fuzzy control as
a universal control tool,” Fuzzy Sets Syst., vol. 80, pp. 71–86, 1996.
J. A. Dickerson and B. Kosko, “Fuzzy function approximation with ellipsoidal rules,” IEEE Trans. Syst., Man, Cybern., vol. 26, pp. 542–560,
1996.
J. J. Buckley, “Sugeno type controllers are universal controllers,” Fuzzy
Sets Syst., vol. 53, pp. 299–303, 1993.
J. L. Castro and M. Delgado, “Fuzzy systems with defuzzification are
universal approximators,” IEEE Trans. Syst., Man, Cybern., vol. 26, pp.
149–152, 1996.
J. L. Castro, “Fuzzy logic controllers are universal approximators,”
IEEE Trans. Syst., Man, Cybern., vol. 25, pp. 629–635, 1995.
L. X. Wang, “Fuzzy systems are universal approximators,” in Proc.
IEEE Int. Conf. Fuzzy Syst., San Diego, CA, Mar. 1992, pp. 1163–1170.
L. X. Wang and J. M. Mendel, “Fuzzy basis functions, universal approximation, and orthogonal least-square learning,” IEEE Trans. Neural Networks, vol. 3, pp. 807–814, 1992.
L. X. Wang, “Universal approximation by hierarchical fuzzy systems,”
Fuzzy Sets Syst., vol. 93, pp. 223–230, 1998.
P. Z. Wang, S. H. Tan, and F. M. Song et al., “Constructive theory for
fuzzy systems,” Fuzzy Sets Syst., vol. 88, pp. 195–203, 1997.
R. Rovatti, “Fuzzy piecewise multi-linear and piecewise linear systems
as universal approximators in Sobolev norms,” IEEE Trans. Fuzzy Syst.,
vol. 6, pp. 235–249, May 1998.
S. Abe and M. S. Lan, “Fuzzy rules extraction directly from numerical
data for function approximation,” IEEE Trans. Syst., Man, Cybern., vol.
25, pp. 119–129, 1995.
X. J. Zeng and M. G. Singh, “Approximation theory of fuzzy systems-SISO case,” IEEE Trans. Fuzzy Syst., vol. 2, pp. 162–176, 1994.
, “Approximation theory of fuzzy systems-MIMO case,” IEEE
Trans. Fuzzy Syst., vol. 3, pp. 219–235, 1995.
, “Approximation accuracy analysis of fuzzy systems as function
approximators,” IEEE Trans. Fuzzy Syst., vol. 4, pp. 44–63, Feb. 1996.
G. G. Lorentz, Approximation of Functions. London, U.K.: Holt, Rinehart, Winston, 1966.
H. Ying, “Necessary conditions for some typical fuzzy systems as universal approximators,” Automat., vol. 33, no. 7, pp. 1333–1338, 1997.
H. Ying, Y. S. Ding, and S. K. Li et al., “Comparison of necessary condition for typical Takagi-Sugeno and Mamdani fuzzy systems as universal
approximators,” IEEE Trans. Syst., Man, Cybern., vol. 29, pp. 508–514,
1999.