Approximation properties of fuzzy systems for smooth ... - IEEE Xplore
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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 33, NO. 2, MARCH 2003
Approximation Properties of Fuzzy Systems for Smooth Functions and Their First-Order Derivative Radhia Hassine, Fakhreddine Karray, Senior Member, IEEE, Adel M. Alimi, Senior Member, IEEE, and Mohamed Selmi
Abstract—The problem of simultaneous approximations of a given function and its derivatives, has been addressed frequently in pure and applied mathematics. In pure mathematics, Bernstein polynomials get their importance from the fact that they provide simultaneous approximation of a function and its derivatives. In neural network theory, feedforward networks were shown to be universal approximators of an unknown function and its derivatives. In this paper, we consider fuzzy logic systems with the membership functions of each input variables are chosen as the translations and dilations of one appropriately fixed function. We prove, by a constructive proof based on discretization of the convolution operator, that under certain conditions made on the input variables membership functions, fuzzy logic systems of Sugeno type are universal approximators of a given function and its derivatives. Index Terms—Beta functions, first-order derivative functions approximation, functions approximation, fuzzy logic systems, Sugeno type fuzzy systems.
I. INTRODUCTION
T
HE UNIVERSAL approximation properties of fuzzy logic systems (FLSs) have been extensively studied in fuzzy logic literature. Several researchers have shown that Mamdani fuzzy logic systems are universal approximators [12], [28]–[30], [33], and [34]. Others have also shown that the Sugeno based fuzzy systems have the universal approximation property [2]–[4], [7], [13], [17], [31], [32]. However, in some applications, it might be necessary to approximate the initial function and its first-order derivatives. This is the case in Jordan’s investigation of robot learning of smooth movement [15] where it is useful to learn an adequate approximation to the jacobian matrix (which is the matrix of the partial derivatives). Another application where we need to approximate the first Manuscript received July 23, 2001; revised February 6, 2002. This work was supported by grants from NSERC-Canada and the General Direction of Scientific Research (DGRST), Tunisia, under the ARUB program. Parts of this paper were developed when the first author was a Visiting Researcher at PAMI Laboratory, University of Waterloo, Waterloo, ON, Canada. This paper was recommended by Associate Editor N. Pal. R. Hassine is with the Department of Mathematics, Faculty of Sciences of Monastir, University of Center, Monastir 5000, Tunisia (e-mail: [email protected]). F. Karray is with the Pattern Analysis and Machine Intelligence Laboratory, Systems Design Engineering Department, University of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail: [email protected]). A. M. Alimi is with the Research Group on Intelligent Machines, Department of Electrical Engineering, University of Sfax, Sfax 3038, Tunisia (e-mail: [email protected]). M. Selmi is with the Laboratory of Physics and Mathematics, Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Sfax 3038, Tunisia (e-mail: [email protected]). Digital Object Identifier 10.1109/TSMCA.2003.811772
derivative of a function is in economics where theoretical considerations lead to hypotheses about the derivative properties of certain functions arising in the theory of the firm and the consumer [9], [24]–[26]. Approximation of derivatives also permits the analysis of the effects of small changes in input variables on output ones. Such analysis was studied in the neural network case [10]. The problem of simultaneous approximation of a given function and its derivatives, is a well known problem in the pure and the applied mathematics fields. In pure mathematics, Bernstein polynomials [8] get their importance from the fact that they provide simultaneous approximation of a function and its derivatives. In neural network theory, feedforward networks were shown to be universal approximators of a given function and its derivatives [14]. The issue of approximating a function and its derivatives by FLS is analytically more complex than that for the neural networks case. In fact FLSs are generally represented as a ratio of linear combination of the input variables membership functions and the sum of the membership functions. In consequence the expression of even the first derivative of a FLS is more complex than the neural network counterpart. To ensure the capability of simultaneous approximations by FLSs, the input variables membership functions should be well chosen. In fact, the choice of the membership functions widely affects the behavior of FLSs. Few research work have been interested in this issue such as that of Kreinovich et al. who considered FLSs with Gaussian input variables membership functions [18]. A more general result has been proved by Landajo et al. [19], who proved that FLSss are universal approximators in the . In this paper we prove that under certain Frechet space hypotheses on the input variables membership functions, FLSs are universal approximators to a given function defined on a and to its first-order derivative. compact subset of The paper is organized as follows. In Section II, we give the analytical representation of the FLS. Some properties pertaining to membership functions construction are then provided. In Section IV, we prove that under certain hypotheses on the membership functions, our constructed FLS of single-input–singleoutput (SISO) type is a universal approximator not only for the given function but also for its first order derivative. This result, is then generalized to the multi-input-single-output (MISO) case. Concluding remarks are then provided. II. GENERAL FORM OF AN FLS MISO FLS can be seen as a function , where is the input space, is the output space. As has been shown by Lee [20], a MIMO fuzzy system can always be
1083-4427/03$17.00 © 2003 IEEE
HASSINE et al.: APPROXIMATION PROPERTIES OF FUZZY SYSTEMS FOR SMOOTH
Fig. 1. Graph of the functions T(x), T(2x
0 1) and T(x=2 + 1).
separated into a group of MISO fuzzy systems, so it is sufficient to study MISO fuzzy systems and then the results concerning MIMO ones easily follow. In this paper, we adopt the Sugeno fuzzy model of zero order with the multiplication operator as its t-norm, then a fuzzy system is given by
(1) where the input variable; the set ; the number of fuzzy rules of the following form: if
is
then
constants in which represent the consequence of each fuzzy rule . linguistic terms characterized by their . membership functions It is clear that in SISO case (1) could be simplified as follows: (2) The rule base consists of If
fuzzy rules of the following form: is
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then
where
are the input variables membership functions and are constants which represent the consequence of each fuzzy rule. III. INPUT MEMBERSHIP FUNCTIONS The choice of the input variables membership functions is quite important as it could affects substantially its output behavior. Many classes of membership functions have been proposed in the literature. These include the triangular functions [22], the normal peak functions [30], the pseudo trapezoidal functions [33], [34], the beta functios [2], [3] and other functions that use translations and dilations of the enumerated functions [21]. As was mentioned in [27] and [21] a number of factors should be considered when choosing the type of membership functions. For example the appropriate membership functions should be intuitively meaningful, easily realizable and able to solve general class of problems. Next, we explain our construction of membership functions in the SISO case. In this paper, the membership functions of each input variable are chosen as the translations and dilations of one appropriately fixed function. The translations and dilations of a function are defined as follows: be a given function defined on , the Definition: Let are defined by the following translations and dilations of relation:
where . is the translation factor, and is the dilation factor. is called the basis function. Fig. 1 shows some translations and dilations of . the triangular membership function As has been mentioned in [21], this choice has a number of advantages. One of them is that if is intuitively meaningful, will be also intuitively meaningful. Anthen all are translations and other advantage, is that since
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Lemma 1: Let be a function satisfying the hypotheses of the previous theorem and let us define the sequence of functions by
then
is a regularizing sequence constructed from . satisfies the following properProof: Indeed
ties: 1) 2) 3)
Fig. 2. Graph of the function A.
dilations of they can change their width and “move” on the real line to be able of representing different linguistic variables. But the main advantage of this construction in our point of view is the flexibility and the ease of propagating properties of the fixed function to all other membership functions. For the MISO case, we first define (3) , then all membership functions will where . This construction represents be of the form the same advantages as for the SISO case. IV. UNIVERSAL APPROXIMATION IN THE SISO CASE
for all
;
with and this is due to the fact that is of compact support so, we can find a strictly positif real number such that supp , in consequence . In order to make the proof natural, we will start with the analysis of how the next theorem can be proved. First, given the function which satisfies the desired properties of the theorem, is constructed as indicated in a regularizing sequence the previous lemma. The essential property of this sequence is with any continuous functhe following: The convolution of tion uniformly approximates the initial function . Let us recall that the convolution of two functions and is the function and defined as follows: denoted by
By the same reason also approximates in the infinity is also continuous because , so norm. Indeed and for sufficiently large n the two quantities can be made less than . These properties can be found in [23]. After that we will discretize the quantity . In fact, this integral is identically null out, so side a compact
Theorem 1: Let be a given function and let be a fixed real number. Then there exists a FLS given by: (4) where
,
,
and
are properly chosen such that
By using the definition of the Riemann integral, the previous quantity can be approximated by which has the same limit as the following expression:
(5) and (6) is a function in satisfying the Where following hypotheses: is of compact support, is even and , and is the closure in of the following . set Before giving the proof of the theorem we need to recall the definition of a regularizing sequence and to prove the following lemma. Definition of a Regularizing Sequence: Let be a normed algebra, a sequence of elements of is said we to be a regularizing sequence if for every element have
This function will represent the FLS we are looking for. By using the triangular inequality, we see that
which can be made less than for sufficiently large . In fact while using fuzzy logic terms, represents the number of fuzzy rules. so to get better approximation we should increase the number of rules. Now let us remark that
HASSINE et al.: APPROXIMATION PROPERTIES OF FUZZY SYSTEMS FOR SMOOTH
Fig. 3. Graph of the functions Ax, A(2x
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0 1) and A(x=2 + 1).
TABLE I RELATIVE APPROXIMATION ERRORS OF DIFFERENT FUNCTIONS BY BETA FLSS AND GAUSS FLSS
Moreover and have compact supports, the same thing . holds for ), so we can find a (In fact such that . real number supp so, Let Let , and by the definition of the Riemann integral we have:
, then
We also have by the same reason
The second term of this sum tends to which is equal to zero because another time the triangular inequality for that
which is equal to is odd. By using and we can see
which can also be made smaller than . So by taking sufficiently large number of rules we the constructed FLS not only approximates the initial function but its first derivative as well. The proof of theorem 1 is given next. be the function to Proof of Theorem 1: Let be a fixed real number. Because be approximated and let is a regularizing sequence constructed from , we can find a strictly positif integer such that for all we have
So . Moreover the convergence is uniform over a compact set. Let
(9) then, for
(
is a fixed rank), we have
(7) and (8) where
denotes the convolution operator.
which is
(10) In consequence, for all
we get (11)
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TABLE II RELATIVE APPROXIMATION ERRORS OF THE DERIVATIVES OF DIFFERENT FUNCTIONS BY THE DERIVATIVES OF BETA FLSS AND THE DERIVATIVES OF GAUSS FLSS
TABLE III PARAMETERS OF THE BETA FLS DESIGNED TO APPROXIMATE THE PARABOLE FUNCTION AND ITS FIRST-ORDER DERIVATIVE
Example 1: The Now we can prove that
approximates
function
. In fact, we have
(12)
which is as N tends to infinity we see that tends to . equal to (This is a general property of the convolution operator.) It is also easy to show that
One can easily verify that for all real number ; 1) is in ; 2) is of compact support which is ; 3) is even; 4) . 5) Figs. 2 and 3 show the graph of the function and some of its translations and dilations variants. Example 2: The Beta function [4]
elsewhere where real numbers. Example 3:
, and are strictly positive
elsewhere then the second term of (12) which is tend to zero.(This is due to the fact that , we have Moreover,
will is odd, and for all ). .
So for sufficiently large . Then
we have
(14)
(15)
where Remark: The function, which was given in Example 1, also presents the two main following properties: , where 1) is a normalized fuzzy set, that is . is a spline function of degree two, so is constructed 2) from polynomials of degree two which are well connected, and we all know that polynomials are the most simple among all functions to manipulate. V. UNIVERSAL APPROXIMATION IN THE MISO CASE
(13) to have
ma It is now sufficient to take a rank the theorem proved. To illustrate the fact that such a function exists, let us give the following examples.
This section deals with the extension of the earlier results to the MISO case. For this reason, we need the following lemma which we can prove in exactly the same manner as lemma 1. and Lemma 2: Let , then is a regularizing sequence in .
HASSINE et al.: APPROXIMATION PROPERTIES OF FUZZY SYSTEMS FOR SMOOTH
TABLE IV PARAMETERS OF THE GAUSS FLS DESIGNED TO APPROXIMATE THE PARABOLE FUNCTION AND ITS FIRST-ORDER DERIVATIVE
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Let . and We also have by the same reason
,
Then, Theorem 2: Let be a given function and be a fixed real number. Then there is a FLS given by
the convergence is uniform over set. Let
. Moreover which is a compact
(21) (16) where and
and , , , ( ) are appropriately chosen such that
then, for inf
(
is a fixed rank) we have (22)
(17) and (18) is a function in satisfying the Where following hypotheses: is of compact support, is even and , where supp is the closure in of the set . be the function to be approxiProof: Let be a fixed real number. because mated and let is a regularizing sequence in , we can we have find a strictly positif integer such that for all
In consequence, for all we get
such that inf
(23) approximates Now, we prove that . We have (24), shown at the bottom of the page), as tends to infinity, we see that inf approximates . In fact it is easy to show that
(19) and (20) , where always denote the convolution for all and are of compact supports, the same operator. Moreover , so we can find a real number thing will be true for such that . supp then, Let
(This is due to the fact that and such that inf For ciently large we have
is odd). is greater than suffi-
(25) This leads to
Let and integral we have:
, , then by the definition of the Riemann (26)
(24)
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TABLE V PARAMETERS OF THE BETA FLS DESIGNED TO APPROXIMATE THE SINE FUNCTION AND ITS FIRST-ORDER DERIVATIVE
VI. SIMULATION RESULTS In this section, we always consider the Sugeno fuzzy model of zero order with fuzzy rules (27) are respectively chosen such that Beta where the functions membership functions, and gauss membership functions. We used a modified version of the supervised gradient descent [6], [11] to tune all the parameters of the fuzzy sets in order to minimize the relative error (28) With the previously designed FLS we computed the relative error between the first derivative of the approximand function and the first derivative of the obtained FLS, that is
TABLE VI PARAMETERS OF THE GAUSS FLS DESIGNED TO APPROXIMATE THE SINE FUNCTION AND ITS FIRST-ORDER DERIVATIVE
(29) are chosen among the folThe approximand functions lowing ones which are the parabola function, the sine function, the breakedsine function and the logarithm function. (30) (31) (32) (33) The simulation results are shown in Tables I and II and they confirm that Beta FLSs [2]–[4] are better than a widely used class of fuzzy logic systems, which are Gauss FLSs with gaussian input variables membership functions. Gauss membership functions depend upon two parameters and , they are defined as follows:
TABLE VII PARAMETERS OF THE BETA FLS DESIGNED TO APPROXIMATE THE BREAKED SINE FUNCTION AND ITS FIRST-ORDER DERIVATIVE
(34) Beta membership functions depend upon four parameters , , and they are defined by the following expression: (35) elsewhere.
It is now sufficient to take all theorem proved.
sufficiently great to have the
The parameters of the designed Beta FLSs and Gauss FLSs are given in Tables III–X where the subscript refers to the rule number and the are the weights of each fuzzy rule . The good performances of Beta FLSs can be explained as follows: Beta functions depend upon four parameters , , , and . Parameters and determine the support of the Beta function which can be translated, shrunk or dilated according to the values of and . and , allow Beta functions to have and asymmetric different shapes which are symmetric if . This flexibility in shape added to the fact that beta if functions are of compact support allow Beta FLSs to be very good function approximators [4].
HASSINE et al.: APPROXIMATION PROPERTIES OF FUZZY SYSTEMS FOR SMOOTH
TABLE VIII PARAMETERS OF THE GAUSS FLS DESIGNED TO APPROXIMATE THE BREAKED SINE FUNCTION AND ITS FIRST-ORDER DERIVATIVE
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ACKNOWLEDGMENT The authors wish to thank Prof. L. A. Zadeh for his advices and the fruitful discussions at the International Conference on Artificial and Computational Intelligence for Decision, Control, and Automation in Engineering and Industrial Applications, Monastir, Tunisia. They also wish to thank Prof. N. Derbel for his continuous help and support. REFERENCES
TABLE IX PARAMETERS OF THE BETA FLS DESIGNED TO APPROXIMATE THE LOGARITHMIC FUNCTION AND ITS FIRST-ORDER DERIVATIVE
TABLE X PARAMETERS OF THE GAUSS FLS DESIGNED TO APPROXIMATE THE LOGARITHMIC FUNCTION AND ITS FIRST-ORDER DERIVATIVE
VII. CONCLUSION In this paper, we have shown that FLSs with appropriately chosen membership functions are universal approximators for a given function and its first order derivative. This result is important not only because it gives more theoretical foundations for the use of FLSs but it also widens their scope of use in a wider range of applications [1], [5], [16]. In fact, in control applications we might be needing to approximate a given function and its first order derivative. Moreover, all membership functions used in the construction of the FLS are the result of translations and dilations of one appropriately chosen fixed membership function, this makes the expression of FLS simpler and more practical to use.
[1] A. M. Alimi, “Evolutionary computation for the recognition of on-line cursive handwriting,” IETE J. Res., vol. 48, no. 5, pp. 385–396, 2002. [2] A. Alimi, R. Hassine, and M. Selmi, “Beta fuzzy logic systems: approximation properties in the SISO case,” Int. J. Appli. Math. Comput. Sci., vol. 10, no. 4, pp. 857–875, 2000. [3] , “Beta fuzzy logic systems: approximation properties in the MIMO case,” Int. J. Appli. Math. Comput. Sci., vol. 13, no. 2, pp. 101–114, 2003. [4] M. A. Alimi, “The beta system: toward a change in our use of neurofuzzy systems,” Int. J. Manage., pp. 15–19, June 2000. [5] C. Aouiti, M. A. Alimi, and A. Maalej et al., “A genetic designed beta basis function neural network for approximating multi-variables functions,” in Artificial Neural Nets and Genetic Algorithms, V. Kurkova et al., Eds. New York: Springer-Verlag, 2001, pp. 383–386. [6] M. J. Box, D. Davies, and W. H. Swann, Non-linear optimization techniques. Edinburgh, U.K.: Oliver & Boyd, 1989. [7] J. J. Buckley, “Sugeno type controllers are universal controllers,” Fuzzy Sets Syst., vol. 52, pp. 299–303, 1993. [8] P. J. Davis, Interpolation and Approximation. London, U.K.: Blaisdell, 1965. [9] I. Elbadawi, A. R. Gallant, and G. Souza, “An elasticity can be estimated without a priori knowledge of functional form,” Econometrica, vol. 51, pp. 1731–1752, 1983. [10] L. Gilstrap and R. Dominy, “A general explanation and interrogation system for neural networks,” in Int. Joint Conf. Neural Networks, Washington, D.C., 1989. [11] A. Grace, Optimization Toolbox For Use With MATLAB. Natick, MA: Math Works, 1994. [12] R. Hartani, T. H. Nguyen, and B. Bouchon-Meunier, “Sur l’approximation universelle des systémes flous,” RAIRO, APII-JESA, vol. 30, no. 5, pp. 645–663, 1996. [13] R. Hassine, M. A. Alimi, and M. Selmi, “What about the best approximation property of beta fuzzy logic systems?,” in New Frontiers in Computational Intelligence and its Applications , M. Mohammadian, Ed. Amsterdam, The Netherlands: IOS, 2000, pp. 62–67. [14] K. Hornik, M. Stinchcombe, and H. White, “Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks,” Neural Networks, vol. 3, pp. 551–560, 1990. [15] M. Jordan, “Generic constraints on under specified target trajectories,” in Proc. 1989 Int. Joint Conf. Neural Networks, vol. I. Piscataway, NJ, 1989, pp. 217–225. [16] F. Karray, “Soft computing techniques for the design of intelligent machines,” in Intelligent Machines: Myths and Realities, C. De Silva, Ed. Boca Raton, FL: CRC, 2000, pp. 184–193. [17] B. Kosko, “Fuzzy systems as universal approximators,” in Proc. IEEE Int. Conf. Fuzzy Syst., San Diego, CA, 1992, pp. 1153–1162. [18] V. Kreinovich, H. T. Nguyen, and Y. Yam, “Fuzzy systems are universal approximators for a smooth function and its derivatives,” Int. J. Intell. Syst., vol. 15, pp. 565–574, 2000. [19] M. Landajo, M. J. Rio, and R. Pérez, “A note on smooth spproximation capabilities of fuzzy systems,” IEEE Trans. Fuzzy Syst., vol. 9, pp. 229–237, Apr. 2001. [20] C. C. Lee, “Fuzzy logic in control systems: fuzzy logic control—part I,” IEEE Trans. Syst., Man, Cybern., vol. 20, pp. 404–418, Mar./Apr. 1990. [21] Z.-H. Mao, Y.-D. Li, and X.-F. Zhang, “Approximation capability of fuzzy systems using translations and dilations of one fixed function as membership function,” IEEE Trans. Fuzzy Syst., vol. 5, pp. 468–473, Aug. 1997. [22] W. Pedrycz, “Why triangular membership functions,” Fuzzy Sets Syst., no. 64, pp. 21–30, 1994.
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[23] L. Serge, Analysis II. Reading, MA: Addison-Wesley, 1969, ch. XIV. [24] A. Ullah, “Non parametric estimation of econometric functionals,” Can. J. Econ., no. 21, pp. 625–658, 1988. [25] H. Varian, Microeconomic Analysis. New York: Norton, 1978. [26] H. D. Vinod and A. Ullah, “Flexible Production Function Estimation by Non-Parametric Kernel Estimators,” Univ. Western Ontario, London, ON, Canada, Dept. Economics, 1985. [27] C. H. Wang, W. Y. Wang, T. T. Lee, and P. S. Tseng, “Fuzzy B-spline membership function and its applications in fuzzy-neural control,” IEEE Trans. Syst. Man, Cybern., vol. 25, pp. 841–851, May 1995. [28] L.-X. Wang, “Fuzzy systems are universal approximators,” in Proc. IEEE Int. Conf. Fuzzy Systems, San Diego, CA, 1992. [29] L.-X. Wang and J. M. Mendel, “Fuzzy basis functions, universal approximations and orthogonal least squares learning,” IEEE Trans. Neural Networks, vol. 3, pp. 807–814, 1992. [30] P.-Z. Wang, S. Tan, F. Song, and P. Liang, “Constructive theory of fuzzy systems,” Fuzzy Sets Syst., no. 88, pp. 195–203, 1997. [31] H. Ying, “Constructing nonlinear variable gain controllers via the Takagi-Sugeno Fuzzy control,” IEEE Trans. Fuzzy Syst., vol. 6, pp. 226–234, May 1998. , “General SISO Takagi-Sugeno fuzzy systems with linear rule con[32] sequent are universal approximators,” IEEE Trans. Fuzzy Syst., vol. 6, pp. 582–587, May 1998. [33] X.-J. Zeng and M. G. Singh, “Approximation theory of fuzzy systems—SISO case,” IEEE Trans. Fuzzy Syst., vol. 2, pp. 162–176, May 1994. , “Approximation theory of fuzzy systems—MIMO case,” IEEE [34] Trans. Fuzzy Syst., vol. 3, pp. 219–235, May 1995.
Fakhreddine Karray (SM’99) received the Ing. Diplome degree in electrical engineering from University of Tunis (ENIT), Tunis, Tunisia in in 1984 and the Ph.D. degree from the University of Illinois, Urbana-Champaign, in 1989 in the area of systems and controls. He was in the faculty of engineering at the University of British Columbia and Lakehead University before joining the University of Waterloo, Waterloo, ON, Canada, in 1997, where he is currently an Associate Professor in the Department of Systems Design Engineering and the Department of Electrical and Computer Engineering and the Associate Director of the UW’s Pattern Analysis and Machine Intelligence Laboratory. His areas of expertise span the fields of intelligent systems design and advanced controls using tools of computational intelligence with application to a wide range of industries in the manufacturing, automotive, and telecommunication sectors. He has published in these areas refereed articles that appeared in journals, textbooks, encyclopedias and conference proceedings. He is also the author of seven provisional patents. He has consulted and advised a number of high tech companies in Canada and the USA and is the cofounder of Intelligent Mechatronics Systems, Inc. and VESTEC, Inc. Dr. Karray has served on several occasions as a member of international program committees of international conferences and symposia and was the Program Chair of the 2002 IEEE International Symposium on Intelligent Control. He is also an Associate Editor and Member of the Editorial Board of three technical journals and IEEE conference proceedings. He is the recipient of three best papers awards, 1998, 1995, and 1993, an IEEE Technical Speaker award, Mexico, 1999 and the Ontario Premier Research Excellence Award in 2000. He is the Chair of the IEEE Control Systems Society Kitchener Waterloo Chapter and an active member of the IEEE Technical Committee on Intelligent Control.
Adel M. Alimi (SM’00) was born in Sfax, Tunisia, in 1966. He received the B.E.E. degree in 1990, and the Ph.D. and H.D.R. degrees both in electrical engineering, in 1995 and 2000, respectively. He is now Associate Professor in electrical and computer engineering at the University of Sfax. His research interest includes applications of intelligent methods, neural networks, fuzzy logic, genetic algorithms to pattern recognition, robotic systems, vision systems, and industrial processes. He focuses his research on intelligent pattern recognition, learning, analysis and intelligent control of large scale complex systems. He is Associate Editor of the Pattern Recognition Letters. He was Guest Editor of special issues of Fuzzy Sets and Systems Journal, Integrated Computer Aided Engineering Journal, Systems Analysis Modeling and Simulations Journal. Dr. Alimi was the General Cochairman of the international conference ACIDCA’2000 that was organized in Monastir, Tunisia. He is a member of IAPR, INNS, and PRS. Radhia Hassine was born in Tunisia in 1971. She received the M.S. degree in mathematics in 1993 from the University of Center, Monastir, Tunisia. She is currently pursuing the Ph.D. degree in applied mathematics with the National School of Engineering of Tunis, Tunis, Tunisia. Since 1997 she has been teaching at the Faculty of Sciences, the University of Center. Her research fields include fuzzy systems, their approximation properties and the comparison of the approximation properties of different families of fuzzy logic systems.
Mohamed Selmi was born in Tunisia in 1952. He received the M.S. degree in the area of mathematical physics and Hamiltonian systems in 1979 from the University of Paris, Paris, France and the Ph.D. degree in 1992, in the area of groups and representation theory. He is currently a Professor of Mathematics at the Faculty of Sciences, University of Sfax, Sfax, Tunisia. His fields of interest include the generalized moment mapping associated to representation theory, the star product and the deformation theory.
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 33, NO. 2, MARCH 2003
Approximation Properties of Fuzzy Systems for Smooth Functions and Their First-Order Derivative Radhia Hassine, Fakhreddine Karray, Senior Member, IEEE, Adel M. Alimi, Senior Member, IEEE, and Mohamed Selmi
Abstract—The problem of simultaneous approximations of a given function and its derivatives, has been addressed frequently in pure and applied mathematics. In pure mathematics, Bernstein polynomials get their importance from the fact that they provide simultaneous approximation of a function and its derivatives. In neural network theory, feedforward networks were shown to be universal approximators of an unknown function and its derivatives. In this paper, we consider fuzzy logic systems with the membership functions of each input variables are chosen as the translations and dilations of one appropriately fixed function. We prove, by a constructive proof based on discretization of the convolution operator, that under certain conditions made on the input variables membership functions, fuzzy logic systems of Sugeno type are universal approximators of a given function and its derivatives. Index Terms—Beta functions, first-order derivative functions approximation, functions approximation, fuzzy logic systems, Sugeno type fuzzy systems.
I. INTRODUCTION
T
HE UNIVERSAL approximation properties of fuzzy logic systems (FLSs) have been extensively studied in fuzzy logic literature. Several researchers have shown that Mamdani fuzzy logic systems are universal approximators [12], [28]–[30], [33], and [34]. Others have also shown that the Sugeno based fuzzy systems have the universal approximation property [2]–[4], [7], [13], [17], [31], [32]. However, in some applications, it might be necessary to approximate the initial function and its first-order derivatives. This is the case in Jordan’s investigation of robot learning of smooth movement [15] where it is useful to learn an adequate approximation to the jacobian matrix (which is the matrix of the partial derivatives). Another application where we need to approximate the first Manuscript received July 23, 2001; revised February 6, 2002. This work was supported by grants from NSERC-Canada and the General Direction of Scientific Research (DGRST), Tunisia, under the ARUB program. Parts of this paper were developed when the first author was a Visiting Researcher at PAMI Laboratory, University of Waterloo, Waterloo, ON, Canada. This paper was recommended by Associate Editor N. Pal. R. Hassine is with the Department of Mathematics, Faculty of Sciences of Monastir, University of Center, Monastir 5000, Tunisia (e-mail: [email protected]). F. Karray is with the Pattern Analysis and Machine Intelligence Laboratory, Systems Design Engineering Department, University of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail: [email protected]). A. M. Alimi is with the Research Group on Intelligent Machines, Department of Electrical Engineering, University of Sfax, Sfax 3038, Tunisia (e-mail: [email protected]). M. Selmi is with the Laboratory of Physics and Mathematics, Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Sfax 3038, Tunisia (e-mail: [email protected]). Digital Object Identifier 10.1109/TSMCA.2003.811772
derivative of a function is in economics where theoretical considerations lead to hypotheses about the derivative properties of certain functions arising in the theory of the firm and the consumer [9], [24]–[26]. Approximation of derivatives also permits the analysis of the effects of small changes in input variables on output ones. Such analysis was studied in the neural network case [10]. The problem of simultaneous approximation of a given function and its derivatives, is a well known problem in the pure and the applied mathematics fields. In pure mathematics, Bernstein polynomials [8] get their importance from the fact that they provide simultaneous approximation of a function and its derivatives. In neural network theory, feedforward networks were shown to be universal approximators of a given function and its derivatives [14]. The issue of approximating a function and its derivatives by FLS is analytically more complex than that for the neural networks case. In fact FLSs are generally represented as a ratio of linear combination of the input variables membership functions and the sum of the membership functions. In consequence the expression of even the first derivative of a FLS is more complex than the neural network counterpart. To ensure the capability of simultaneous approximations by FLSs, the input variables membership functions should be well chosen. In fact, the choice of the membership functions widely affects the behavior of FLSs. Few research work have been interested in this issue such as that of Kreinovich et al. who considered FLSs with Gaussian input variables membership functions [18]. A more general result has been proved by Landajo et al. [19], who proved that FLSss are universal approximators in the . In this paper we prove that under certain Frechet space hypotheses on the input variables membership functions, FLSs are universal approximators to a given function defined on a and to its first-order derivative. compact subset of The paper is organized as follows. In Section II, we give the analytical representation of the FLS. Some properties pertaining to membership functions construction are then provided. In Section IV, we prove that under certain hypotheses on the membership functions, our constructed FLS of single-input–singleoutput (SISO) type is a universal approximator not only for the given function but also for its first order derivative. This result, is then generalized to the multi-input-single-output (MISO) case. Concluding remarks are then provided. II. GENERAL FORM OF AN FLS MISO FLS can be seen as a function , where is the input space, is the output space. As has been shown by Lee [20], a MIMO fuzzy system can always be
1083-4427/03$17.00 © 2003 IEEE
HASSINE et al.: APPROXIMATION PROPERTIES OF FUZZY SYSTEMS FOR SMOOTH
Fig. 1. Graph of the functions T(x), T(2x
0 1) and T(x=2 + 1).
separated into a group of MISO fuzzy systems, so it is sufficient to study MISO fuzzy systems and then the results concerning MIMO ones easily follow. In this paper, we adopt the Sugeno fuzzy model of zero order with the multiplication operator as its t-norm, then a fuzzy system is given by
(1) where the input variable; the set ; the number of fuzzy rules of the following form: if
is
then
constants in which represent the consequence of each fuzzy rule . linguistic terms characterized by their . membership functions It is clear that in SISO case (1) could be simplified as follows: (2) The rule base consists of If
fuzzy rules of the following form: is
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then
where
are the input variables membership functions and are constants which represent the consequence of each fuzzy rule. III. INPUT MEMBERSHIP FUNCTIONS The choice of the input variables membership functions is quite important as it could affects substantially its output behavior. Many classes of membership functions have been proposed in the literature. These include the triangular functions [22], the normal peak functions [30], the pseudo trapezoidal functions [33], [34], the beta functios [2], [3] and other functions that use translations and dilations of the enumerated functions [21]. As was mentioned in [27] and [21] a number of factors should be considered when choosing the type of membership functions. For example the appropriate membership functions should be intuitively meaningful, easily realizable and able to solve general class of problems. Next, we explain our construction of membership functions in the SISO case. In this paper, the membership functions of each input variable are chosen as the translations and dilations of one appropriately fixed function. The translations and dilations of a function are defined as follows: be a given function defined on , the Definition: Let are defined by the following translations and dilations of relation:
where . is the translation factor, and is the dilation factor. is called the basis function. Fig. 1 shows some translations and dilations of . the triangular membership function As has been mentioned in [21], this choice has a number of advantages. One of them is that if is intuitively meaningful, will be also intuitively meaningful. Anthen all are translations and other advantage, is that since
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Lemma 1: Let be a function satisfying the hypotheses of the previous theorem and let us define the sequence of functions by
then
is a regularizing sequence constructed from . satisfies the following properProof: Indeed
ties: 1) 2) 3)
Fig. 2. Graph of the function A.
dilations of they can change their width and “move” on the real line to be able of representing different linguistic variables. But the main advantage of this construction in our point of view is the flexibility and the ease of propagating properties of the fixed function to all other membership functions. For the MISO case, we first define (3) , then all membership functions will where . This construction represents be of the form the same advantages as for the SISO case. IV. UNIVERSAL APPROXIMATION IN THE SISO CASE
for all
;
with and this is due to the fact that is of compact support so, we can find a strictly positif real number such that supp , in consequence . In order to make the proof natural, we will start with the analysis of how the next theorem can be proved. First, given the function which satisfies the desired properties of the theorem, is constructed as indicated in a regularizing sequence the previous lemma. The essential property of this sequence is with any continuous functhe following: The convolution of tion uniformly approximates the initial function . Let us recall that the convolution of two functions and is the function and defined as follows: denoted by
By the same reason also approximates in the infinity is also continuous because , so norm. Indeed and for sufficiently large n the two quantities can be made less than . These properties can be found in [23]. After that we will discretize the quantity . In fact, this integral is identically null out, so side a compact
Theorem 1: Let be a given function and let be a fixed real number. Then there exists a FLS given by: (4) where
,
,
and
are properly chosen such that
By using the definition of the Riemann integral, the previous quantity can be approximated by which has the same limit as the following expression:
(5) and (6) is a function in satisfying the Where following hypotheses: is of compact support, is even and , and is the closure in of the following . set Before giving the proof of the theorem we need to recall the definition of a regularizing sequence and to prove the following lemma. Definition of a Regularizing Sequence: Let be a normed algebra, a sequence of elements of is said we to be a regularizing sequence if for every element have
This function will represent the FLS we are looking for. By using the triangular inequality, we see that
which can be made less than for sufficiently large . In fact while using fuzzy logic terms, represents the number of fuzzy rules. so to get better approximation we should increase the number of rules. Now let us remark that
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Fig. 3. Graph of the functions Ax, A(2x
163
0 1) and A(x=2 + 1).
TABLE I RELATIVE APPROXIMATION ERRORS OF DIFFERENT FUNCTIONS BY BETA FLSS AND GAUSS FLSS
Moreover and have compact supports, the same thing . holds for ), so we can find a (In fact such that . real number supp so, Let Let , and by the definition of the Riemann integral we have:
, then
We also have by the same reason
The second term of this sum tends to which is equal to zero because another time the triangular inequality for that
which is equal to is odd. By using and we can see
which can also be made smaller than . So by taking sufficiently large number of rules we the constructed FLS not only approximates the initial function but its first derivative as well. The proof of theorem 1 is given next. be the function to Proof of Theorem 1: Let be a fixed real number. Because be approximated and let is a regularizing sequence constructed from , we can find a strictly positif integer such that for all we have
So . Moreover the convergence is uniform over a compact set. Let
(9) then, for
(
is a fixed rank), we have
(7) and (8) where
denotes the convolution operator.
which is
(10) In consequence, for all
we get (11)
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TABLE II RELATIVE APPROXIMATION ERRORS OF THE DERIVATIVES OF DIFFERENT FUNCTIONS BY THE DERIVATIVES OF BETA FLSS AND THE DERIVATIVES OF GAUSS FLSS
TABLE III PARAMETERS OF THE BETA FLS DESIGNED TO APPROXIMATE THE PARABOLE FUNCTION AND ITS FIRST-ORDER DERIVATIVE
Example 1: The Now we can prove that
approximates
function
. In fact, we have
(12)
which is as N tends to infinity we see that tends to . equal to (This is a general property of the convolution operator.) It is also easy to show that
One can easily verify that for all real number ; 1) is in ; 2) is of compact support which is ; 3) is even; 4) . 5) Figs. 2 and 3 show the graph of the function and some of its translations and dilations variants. Example 2: The Beta function [4]
elsewhere where real numbers. Example 3:
, and are strictly positive
elsewhere then the second term of (12) which is tend to zero.(This is due to the fact that , we have Moreover,
will is odd, and for all ). .
So for sufficiently large . Then
we have
(14)
(15)
where Remark: The function, which was given in Example 1, also presents the two main following properties: , where 1) is a normalized fuzzy set, that is . is a spline function of degree two, so is constructed 2) from polynomials of degree two which are well connected, and we all know that polynomials are the most simple among all functions to manipulate. V. UNIVERSAL APPROXIMATION IN THE MISO CASE
(13) to have
ma It is now sufficient to take a rank the theorem proved. To illustrate the fact that such a function exists, let us give the following examples.
This section deals with the extension of the earlier results to the MISO case. For this reason, we need the following lemma which we can prove in exactly the same manner as lemma 1. and Lemma 2: Let , then is a regularizing sequence in .
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TABLE IV PARAMETERS OF THE GAUSS FLS DESIGNED TO APPROXIMATE THE PARABOLE FUNCTION AND ITS FIRST-ORDER DERIVATIVE
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Let . and We also have by the same reason
,
Then, Theorem 2: Let be a given function and be a fixed real number. Then there is a FLS given by
the convergence is uniform over set. Let
. Moreover which is a compact
(21) (16) where and
and , , , ( ) are appropriately chosen such that
then, for inf
(
is a fixed rank) we have (22)
(17) and (18) is a function in satisfying the Where following hypotheses: is of compact support, is even and , where supp is the closure in of the set . be the function to be approxiProof: Let be a fixed real number. because mated and let is a regularizing sequence in , we can we have find a strictly positif integer such that for all
In consequence, for all we get
such that inf
(23) approximates Now, we prove that . We have (24), shown at the bottom of the page), as tends to infinity, we see that inf approximates . In fact it is easy to show that
(19) and (20) , where always denote the convolution for all and are of compact supports, the same operator. Moreover , so we can find a real number thing will be true for such that . supp then, Let
(This is due to the fact that and such that inf For ciently large we have
is odd). is greater than suffi-
(25) This leads to
Let and integral we have:
, , then by the definition of the Riemann (26)
(24)
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TABLE V PARAMETERS OF THE BETA FLS DESIGNED TO APPROXIMATE THE SINE FUNCTION AND ITS FIRST-ORDER DERIVATIVE
VI. SIMULATION RESULTS In this section, we always consider the Sugeno fuzzy model of zero order with fuzzy rules (27) are respectively chosen such that Beta where the functions membership functions, and gauss membership functions. We used a modified version of the supervised gradient descent [6], [11] to tune all the parameters of the fuzzy sets in order to minimize the relative error (28) With the previously designed FLS we computed the relative error between the first derivative of the approximand function and the first derivative of the obtained FLS, that is
TABLE VI PARAMETERS OF THE GAUSS FLS DESIGNED TO APPROXIMATE THE SINE FUNCTION AND ITS FIRST-ORDER DERIVATIVE
(29) are chosen among the folThe approximand functions lowing ones which are the parabola function, the sine function, the breakedsine function and the logarithm function. (30) (31) (32) (33) The simulation results are shown in Tables I and II and they confirm that Beta FLSs [2]–[4] are better than a widely used class of fuzzy logic systems, which are Gauss FLSs with gaussian input variables membership functions. Gauss membership functions depend upon two parameters and , they are defined as follows:
TABLE VII PARAMETERS OF THE BETA FLS DESIGNED TO APPROXIMATE THE BREAKED SINE FUNCTION AND ITS FIRST-ORDER DERIVATIVE
(34) Beta membership functions depend upon four parameters , , and they are defined by the following expression: (35) elsewhere.
It is now sufficient to take all theorem proved.
sufficiently great to have the
The parameters of the designed Beta FLSs and Gauss FLSs are given in Tables III–X where the subscript refers to the rule number and the are the weights of each fuzzy rule . The good performances of Beta FLSs can be explained as follows: Beta functions depend upon four parameters , , , and . Parameters and determine the support of the Beta function which can be translated, shrunk or dilated according to the values of and . and , allow Beta functions to have and asymmetric different shapes which are symmetric if . This flexibility in shape added to the fact that beta if functions are of compact support allow Beta FLSs to be very good function approximators [4].
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TABLE VIII PARAMETERS OF THE GAUSS FLS DESIGNED TO APPROXIMATE THE BREAKED SINE FUNCTION AND ITS FIRST-ORDER DERIVATIVE
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ACKNOWLEDGMENT The authors wish to thank Prof. L. A. Zadeh for his advices and the fruitful discussions at the International Conference on Artificial and Computational Intelligence for Decision, Control, and Automation in Engineering and Industrial Applications, Monastir, Tunisia. They also wish to thank Prof. N. Derbel for his continuous help and support. REFERENCES
TABLE IX PARAMETERS OF THE BETA FLS DESIGNED TO APPROXIMATE THE LOGARITHMIC FUNCTION AND ITS FIRST-ORDER DERIVATIVE
TABLE X PARAMETERS OF THE GAUSS FLS DESIGNED TO APPROXIMATE THE LOGARITHMIC FUNCTION AND ITS FIRST-ORDER DERIVATIVE
VII. CONCLUSION In this paper, we have shown that FLSs with appropriately chosen membership functions are universal approximators for a given function and its first order derivative. This result is important not only because it gives more theoretical foundations for the use of FLSs but it also widens their scope of use in a wider range of applications [1], [5], [16]. In fact, in control applications we might be needing to approximate a given function and its first order derivative. Moreover, all membership functions used in the construction of the FLS are the result of translations and dilations of one appropriately chosen fixed membership function, this makes the expression of FLS simpler and more practical to use.
[1] A. M. Alimi, “Evolutionary computation for the recognition of on-line cursive handwriting,” IETE J. Res., vol. 48, no. 5, pp. 385–396, 2002. [2] A. Alimi, R. Hassine, and M. Selmi, “Beta fuzzy logic systems: approximation properties in the SISO case,” Int. J. Appli. Math. Comput. Sci., vol. 10, no. 4, pp. 857–875, 2000. [3] , “Beta fuzzy logic systems: approximation properties in the MIMO case,” Int. J. Appli. Math. Comput. Sci., vol. 13, no. 2, pp. 101–114, 2003. [4] M. A. Alimi, “The beta system: toward a change in our use of neurofuzzy systems,” Int. J. Manage., pp. 15–19, June 2000. [5] C. Aouiti, M. A. Alimi, and A. Maalej et al., “A genetic designed beta basis function neural network for approximating multi-variables functions,” in Artificial Neural Nets and Genetic Algorithms, V. Kurkova et al., Eds. New York: Springer-Verlag, 2001, pp. 383–386. [6] M. J. Box, D. Davies, and W. H. Swann, Non-linear optimization techniques. Edinburgh, U.K.: Oliver & Boyd, 1989. [7] J. J. Buckley, “Sugeno type controllers are universal controllers,” Fuzzy Sets Syst., vol. 52, pp. 299–303, 1993. [8] P. J. Davis, Interpolation and Approximation. London, U.K.: Blaisdell, 1965. [9] I. Elbadawi, A. R. Gallant, and G. Souza, “An elasticity can be estimated without a priori knowledge of functional form,” Econometrica, vol. 51, pp. 1731–1752, 1983. [10] L. Gilstrap and R. Dominy, “A general explanation and interrogation system for neural networks,” in Int. Joint Conf. Neural Networks, Washington, D.C., 1989. [11] A. Grace, Optimization Toolbox For Use With MATLAB. Natick, MA: Math Works, 1994. [12] R. Hartani, T. H. Nguyen, and B. Bouchon-Meunier, “Sur l’approximation universelle des systémes flous,” RAIRO, APII-JESA, vol. 30, no. 5, pp. 645–663, 1996. [13] R. Hassine, M. A. Alimi, and M. Selmi, “What about the best approximation property of beta fuzzy logic systems?,” in New Frontiers in Computational Intelligence and its Applications , M. Mohammadian, Ed. Amsterdam, The Netherlands: IOS, 2000, pp. 62–67. [14] K. Hornik, M. Stinchcombe, and H. White, “Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks,” Neural Networks, vol. 3, pp. 551–560, 1990. [15] M. Jordan, “Generic constraints on under specified target trajectories,” in Proc. 1989 Int. Joint Conf. Neural Networks, vol. I. Piscataway, NJ, 1989, pp. 217–225. [16] F. Karray, “Soft computing techniques for the design of intelligent machines,” in Intelligent Machines: Myths and Realities, C. De Silva, Ed. Boca Raton, FL: CRC, 2000, pp. 184–193. [17] B. Kosko, “Fuzzy systems as universal approximators,” in Proc. IEEE Int. Conf. Fuzzy Syst., San Diego, CA, 1992, pp. 1153–1162. [18] V. Kreinovich, H. T. Nguyen, and Y. Yam, “Fuzzy systems are universal approximators for a smooth function and its derivatives,” Int. J. Intell. Syst., vol. 15, pp. 565–574, 2000. [19] M. Landajo, M. J. Rio, and R. Pérez, “A note on smooth spproximation capabilities of fuzzy systems,” IEEE Trans. Fuzzy Syst., vol. 9, pp. 229–237, Apr. 2001. [20] C. C. Lee, “Fuzzy logic in control systems: fuzzy logic control—part I,” IEEE Trans. Syst., Man, Cybern., vol. 20, pp. 404–418, Mar./Apr. 1990. [21] Z.-H. Mao, Y.-D. Li, and X.-F. Zhang, “Approximation capability of fuzzy systems using translations and dilations of one fixed function as membership function,” IEEE Trans. Fuzzy Syst., vol. 5, pp. 468–473, Aug. 1997. [22] W. Pedrycz, “Why triangular membership functions,” Fuzzy Sets Syst., no. 64, pp. 21–30, 1994.
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[23] L. Serge, Analysis II. Reading, MA: Addison-Wesley, 1969, ch. XIV. [24] A. Ullah, “Non parametric estimation of econometric functionals,” Can. J. Econ., no. 21, pp. 625–658, 1988. [25] H. Varian, Microeconomic Analysis. New York: Norton, 1978. [26] H. D. Vinod and A. Ullah, “Flexible Production Function Estimation by Non-Parametric Kernel Estimators,” Univ. Western Ontario, London, ON, Canada, Dept. Economics, 1985. [27] C. H. Wang, W. Y. Wang, T. T. Lee, and P. S. Tseng, “Fuzzy B-spline membership function and its applications in fuzzy-neural control,” IEEE Trans. Syst. Man, Cybern., vol. 25, pp. 841–851, May 1995. [28] L.-X. Wang, “Fuzzy systems are universal approximators,” in Proc. IEEE Int. Conf. Fuzzy Systems, San Diego, CA, 1992. [29] L.-X. Wang and J. M. Mendel, “Fuzzy basis functions, universal approximations and orthogonal least squares learning,” IEEE Trans. Neural Networks, vol. 3, pp. 807–814, 1992. [30] P.-Z. Wang, S. Tan, F. Song, and P. Liang, “Constructive theory of fuzzy systems,” Fuzzy Sets Syst., no. 88, pp. 195–203, 1997. [31] H. Ying, “Constructing nonlinear variable gain controllers via the Takagi-Sugeno Fuzzy control,” IEEE Trans. Fuzzy Syst., vol. 6, pp. 226–234, May 1998. , “General SISO Takagi-Sugeno fuzzy systems with linear rule con[32] sequent are universal approximators,” IEEE Trans. Fuzzy Syst., vol. 6, pp. 582–587, May 1998. [33] X.-J. Zeng and M. G. Singh, “Approximation theory of fuzzy systems—SISO case,” IEEE Trans. Fuzzy Syst., vol. 2, pp. 162–176, May 1994. , “Approximation theory of fuzzy systems—MIMO case,” IEEE [34] Trans. Fuzzy Syst., vol. 3, pp. 219–235, May 1995.
Fakhreddine Karray (SM’99) received the Ing. Diplome degree in electrical engineering from University of Tunis (ENIT), Tunis, Tunisia in in 1984 and the Ph.D. degree from the University of Illinois, Urbana-Champaign, in 1989 in the area of systems and controls. He was in the faculty of engineering at the University of British Columbia and Lakehead University before joining the University of Waterloo, Waterloo, ON, Canada, in 1997, where he is currently an Associate Professor in the Department of Systems Design Engineering and the Department of Electrical and Computer Engineering and the Associate Director of the UW’s Pattern Analysis and Machine Intelligence Laboratory. His areas of expertise span the fields of intelligent systems design and advanced controls using tools of computational intelligence with application to a wide range of industries in the manufacturing, automotive, and telecommunication sectors. He has published in these areas refereed articles that appeared in journals, textbooks, encyclopedias and conference proceedings. He is also the author of seven provisional patents. He has consulted and advised a number of high tech companies in Canada and the USA and is the cofounder of Intelligent Mechatronics Systems, Inc. and VESTEC, Inc. Dr. Karray has served on several occasions as a member of international program committees of international conferences and symposia and was the Program Chair of the 2002 IEEE International Symposium on Intelligent Control. He is also an Associate Editor and Member of the Editorial Board of three technical journals and IEEE conference proceedings. He is the recipient of three best papers awards, 1998, 1995, and 1993, an IEEE Technical Speaker award, Mexico, 1999 and the Ontario Premier Research Excellence Award in 2000. He is the Chair of the IEEE Control Systems Society Kitchener Waterloo Chapter and an active member of the IEEE Technical Committee on Intelligent Control.
Adel M. Alimi (SM’00) was born in Sfax, Tunisia, in 1966. He received the B.E.E. degree in 1990, and the Ph.D. and H.D.R. degrees both in electrical engineering, in 1995 and 2000, respectively. He is now Associate Professor in electrical and computer engineering at the University of Sfax. His research interest includes applications of intelligent methods, neural networks, fuzzy logic, genetic algorithms to pattern recognition, robotic systems, vision systems, and industrial processes. He focuses his research on intelligent pattern recognition, learning, analysis and intelligent control of large scale complex systems. He is Associate Editor of the Pattern Recognition Letters. He was Guest Editor of special issues of Fuzzy Sets and Systems Journal, Integrated Computer Aided Engineering Journal, Systems Analysis Modeling and Simulations Journal. Dr. Alimi was the General Cochairman of the international conference ACIDCA’2000 that was organized in Monastir, Tunisia. He is a member of IAPR, INNS, and PRS. Radhia Hassine was born in Tunisia in 1971. She received the M.S. degree in mathematics in 1993 from the University of Center, Monastir, Tunisia. She is currently pursuing the Ph.D. degree in applied mathematics with the National School of Engineering of Tunis, Tunis, Tunisia. Since 1997 she has been teaching at the Faculty of Sciences, the University of Center. Her research fields include fuzzy systems, their approximation properties and the comparison of the approximation properties of different families of fuzzy logic systems.
Mohamed Selmi was born in Tunisia in 1952. He received the M.S. degree in the area of mathematical physics and Hamiltonian systems in 1979 from the University of Paris, Paris, France and the Ph.D. degree in 1992, in the area of groups and representation theory. He is currently a Professor of Mathematics at the Faculty of Sciences, University of Sfax, Sfax, Tunisia. His fields of interest include the generalized moment mapping associated to representation theory, the star product and the deformation theory.
