A comparative study on sufficient conditions for takagi-sugeno fuzzy ...
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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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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 .
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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.
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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).
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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.
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[9] [10]
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[14] [15] [16]
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, “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.
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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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 .
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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.
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Utilizing Taylor’s formula with Lagrange remainder, we have
where the Lagrange remainder
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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).
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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
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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.
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