Approximation Properties of Fuzzy Systems for Smooth Functions and Their First-Order Derivative

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1 160 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 Their First-Order Derivative Radhia Hassine, Fakhreddine Karray, Senior Member, IEEE, Adel M Alimi, Senior Member, IEEE, Mohamed Selmi Abstract The problem of simultaneous approximations of a given function its derivatives, has been addressed frequently in pure applied mathematics In pure mathematics, Bernstein polynomials get their importance from the fact that they provide simultaneous approximation of a function its derivatives In neural network theory, feedforward networks were shown to be universal approximators of an unknown function its derivatives In this paper, we consider fuzzy logic systems with the membership functions of each input variables are chosen as the translations 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 its derivatives Index Terms Beta functions, first-order derivative functions approximation, functions approximation, fuzzy logic systems, Sugeno type fuzzy systems I INTRODUCTION THE 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], [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 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 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 ( RadhiaHassine@fsmrnutn) F Karray is with the Pattern Analysis Machine Intelligence Laboratory, Systems Design Engineering Department, University of Waterloo, Waterloo, ON N2L 3G1, Canada ( karray@uwaterlooca) A M Alimi is with the Research Group on Intelligent Machines, Department of Electrical Engineering, University of Sfax, Sfax 3038, Tunisia ( AdelAlimi@ieeeorg) M Selmi is with the Laboratory of Physics Mathematics, Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Sfax 3038, Tunisia ( MohamedSelmi@fssrnutn) Digital Object Identifier /TSMCA 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 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 its derivatives, is a well known problem in the pure the applied mathematics fields In pure mathematics, Bernstein polynomials [8] get their importance from the fact that they provide simultaneous approximation of a function its derivatives In neural network theory, feedforward networks were shown to be universal approximators of a given function its derivatives [14] The issue of approximating a function 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 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 Lajo et al [19], who proved that FLSss are universal approximators in the Frechet space In this paper we prove that under certain hypotheses on the input variables membership functions, FLSs are universal approximators to a given function defined on a compact subset of to its first-order derivative 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 /03$ IEEE

2 HASSINE et al: APPROXIMATION PROPERTIES OF FUZZY SYSTEMS FOR SMOOTH 161 Fig 1 Graph of the functions T(x), T(2x 0 1) T(x=2 +1) separated into a group of MISO fuzzy systems, so it is sufficient to study MISO fuzzy systems 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 where the input variable; (1) 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: The rule base consists of (2) fuzzy rules of the following form: If is then where are the input variables membership functions 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] other functions that use translations dilations of the enumerated functions [21] As was mentioned in [27] [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 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 dilations of one appropriately fixed function The translations dilations of a function are defined as follows: Definition: Let be a given function defined on, the translations dilations of are defined by the following relation: where is the translation factor, is the dilation factor is called the basis function Fig 1 shows some translations 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, then all will be also intuitively meaningful Another advantage, is that since are translations

3 162 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART A: SYSTEMS AND HUMANS, VOL 33, NO 2, MARCH 2003 Lemma 1: Let be a function satisfying the hypotheses of the previous theorem let us define the sequence of functions by Fig 2 Graph of the function A dilations of they can change their width 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 the ease of propagating properties of the fixed function to all other membership functions For the MISO case, we first define then is a regularizing sequence constructed from Proof: Indeed satisfies the following properties: 1) for all ; 2) 3) with 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, a regularizing sequence is constructed as indicated in the previous lemma The essential property of this sequence is the following: The convolution of with any continuous function uniformly approximates the initial function Let us recall that the convolution of two functions is the function denoted by defined as follows: where, then all membership functions will be of the form This construction represents the same advantages as for the SISO case IV UNIVERSAL APPROXIMATION IN THE SISO CASE Theorem 1: Let be a given function let be a fixed real number Then there exists a FLS given by: where,, are properly chosen such that Where is a function in satisfying the following hypotheses: is of compact support, is even, 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 to prove the following lemma Definition of a Regularizing Sequence: Let be a normed algebra, a sequence of elements of is said to be a regularizing sequence if for every element we have (3) (4) (5) (6) By the same reason also approximates in the infinity norm Indeed is also continuous because, so 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 outside a compact,so By using the definition of the Riemann integral, the previous quantity can be approximated by which has the same limit as the following expression: 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

4 HASSINE et al: APPROXIMATION PROPERTIES OF FUZZY SYSTEMS FOR SMOOTH 163 Fig 3 Graph of the functions Ax, A(2x 0 1) A(x=2 +1) TABLE I RELATIVE APPROXIMATION ERRORS OF DIFFERENT FUNCTIONS BY BETA FLSS AND GAUSS FLSS Moreover have compact supports, the same thing holds for (In fact ), so we can find a real number such that Let supp so, Let,, then by the definition of the Riemann integral we have: We also have by the same reason The second term of this sum tends to which is equal to which is equal to zero because is odd By using another time the triangular inequality for we can see that 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 Proof of Theorem 1: Let be the function to be approximated let be a fixed real number Because 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 then, for ( is a fixed rank), we have which is (9) (7) (8) In consequence, for all we get (10) where denotes the convolution operator (11)

5 164 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART A: SYSTEMS AND HUMANS, VOL 33, NO 2, MARCH 2003 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 function Now we can prove that approximates In fact, we have (12) as N tends to infinity we see that tends to which is equal to (This is a general property of the convolution operator) It is also easy to show that One can easily verify that 1) for all real number ; 2) is in ; 3) is of compact support which is ; 4) is even; 5) Figs 2 3 show the graph of the function some of its translations dilations variants Example 2: The Beta function [4] elsewhere (14) where, are strictly positive real numbers Example 3: elsewhere (15) then the second term of (12) which is will tend to zero(this is due to the fact that is odd, for all,wehave ) Moreover, So for sufficiently large we have Then (13) It is now sufficient to take a rank ma to have the theorem proved To illustrate the fact that such a function exists, let us give the following examples where Remark: The function, which was given in Example 1, also presents the two main following properties: 1) is a normalized fuzzy set, that is, where 2) is a spline function of degree two, so is constructed from polynomials of degree two which are well connected, we all know that polynomials are the most simple among all functions to manipulate V UNIVERSAL APPROXIMATION IN THE MISO CASE 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 Lemma 2: Let, then is a regularizing sequence in

6 HASSINE et al: APPROXIMATION PROPERTIES OF FUZZY SYSTEMS FOR SMOOTH 165 TABLE IV PARAMETERS OF THE GAUSS FLS DESIGNED TO APPROXIMATE THE PARABOLE FUNCTION AND ITS FIRST-ORDER DERIVATIVE Let, We also have by the same reason Theorem 2: Let be a given function be a fixed real number Then there is a FLS given by Then, the convergence is uniform over set Let Moreover which is a compact where,,, ( ) are appropriately chosen such that (16) (17) (18) then, for inf ( is a fixed rank) we have In consequence, for all such that inf we get (21) (22) Where is a function in satisfying the following hypotheses: is of compact support, is even, where supp is the closure in of the set Proof: Let be the function to be approximated let be a fixed real number because is a regularizing sequence in, we can find a strictly positif integer such that for all we have (19) (20) for all, where always denote the convolution operator Moreover are of compact supports, the same thing will be true for, so we can find a real number such that Let supp then, Let,, then by the definition of the Riemann integral we have: (23) Now, we prove that approximates We have (24), shown at the bottom of the page), as inf tends to infinity, we see that approximates In fact it is easy to show that (This is due to the fact that is odd) For such that inf is greater than sufficiently large we have This leads to (25) (26) (24)

7 166 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART A: SYSTEMS AND HUMANS, VOL 33, NO 2, MARCH 2003 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) where the functions are respectively chosen such that Beta membership functions, 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) TABLE VI PARAMETERS OF THE GAUSS FLS DESIGNED TO APPROXIMATE THE SINE FUNCTION AND ITS FIRST-ORDER DERIVATIVE With the previously designed FLS we computed the relative error between the first derivative of the approxim function the first derivative of the obtained FLS, that is (29) The approxim functions are chosen among the following ones which are the parabola function, the sine function, the breakedsine function the logarithm function (30) (31) (32) (33) TABLE VII PARAMETERS OF THE BETA FLS DESIGNED TO APPROXIMATE THE BREAKED SINE FUNCTION AND ITS FIRST-ORDER DERIVATIVE The simulation results are shown in Tables I II 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, they are defined as follows: (34) Beta membership functions depend upon four parameters,, they are defined by the following expression: It is now sufficient to take all theorem proved sufficiently great to have the (35) elsewhere The parameters of the designed Beta FLSs Gauss FLSs are given in Tables III X where the subscript refers to the rule number 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,,, Parameters determine the support of the Beta function which can be translated, shrunk or dilated according to the values of, allow Beta functions to have different shapes which are symmetric if asymmetric if This flexibility in shape added to the fact that beta functions are of compact support allow Beta FLSs to be very good function approximators [4]

8 HASSINE et al: APPROXIMATION PROPERTIES OF FUZZY SYSTEMS FOR SMOOTH 167 TABLE VIII PARAMETERS OF THE GAUSS FLS DESIGNED TO APPROXIMATE THE BREAKED SINE FUNCTION AND ITS FIRST-ORDER DERIVATIVE ACKNOWLEDGMENT The authors wish to thank Prof L A Zadeh for his advices the fruitful discussions at the International Conference on Artificial Computational Intelligence for Decision, Control, Automation in Engineering Industrial Applications, Monastir, Tunisia They also wish to thank Prof N Derbel for his continuous help 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 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 its first order derivative Moreover, all membership functions used in the construction of the FLS are the result of translations dilations of one appropriately chosen fixed membership function, this makes the expression of FLS simpler more practical to use [1] A M Alimi, Evolutionary computation for the recognition of on-line cursive hwriting, IETE J Res, vol 48, no 5, pp , 2002 [2] A Alimi, R Hassine, M Selmi, Beta fuzzy logic systems: approximation properties in the SISO case, Int J Appli Math Comput Sci, vol 10, no 4, pp , 2000 [3], Beta fuzzy logic systems: approximation properties in the MIMO case, Int J Appli Math Comput Sci, vol 13, no 2, pp , 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, A Maalej et al, A genetic designed beta basis function neural network for approximating multi-variables functions, in Artificial Neural Nets Genetic Algorithms, V Kurkova et al, Eds New York: Springer-Verlag, 2001, pp [6] M J Box, D Davies, W H Swann, Non-linear optimization techniques Edinburgh, UK: Oliver & Boyd, 1989 [7] J J Buckley, Sugeno type controllers are universal controllers, Fuzzy Sets Syst, vol 52, pp , 1993 [8] P J Davis, Interpolation Approximation London, UK: Blaisdell, 1965 [9] I Elbadawi, A R Gallant, G Souza, An elasticity can be estimated without a priori knowledge of functional form, Econometrica, vol 51, pp , 1983 [10] L Gilstrap R Dominy, A general explanation interrogation system for neural networks, in Int Joint Conf Neural Networks, Washington, DC, 1989 [11] A Grace, Optimization Toolbox For Use With MATLAB Natick, MA: Math Works, 1994 [12] R Hartani, T H Nguyen, B Bouchon-Meunier, Sur l approximation universelle des systémes flous, RAIRO, APII-JESA, vol 30, no 5, pp , 1996 [13] R Hassine, M A Alimi, M Selmi, What about the best approximation property of beta fuzzy logic systems?, in New Frontiers in Computational Intelligence its Applications, M Mohammadian, Ed Amsterdam, The Netherls: IOS, 2000, pp [14] K Hornik, M Stinchcombe, H White, Universal approximation of an unknown mapping its derivatives using multilayer feedforward networks, Neural Networks, vol 3, pp , 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 [16] F Karray, Soft computing techniques for the design of intelligent machines, in Intelligent Machines: Myths Realities, C De Silva, Ed Boca Raton, FL: CRC, 2000, pp [17] B Kosko, Fuzzy systems as universal approximators, in Proc IEEE Int Conf Fuzzy Syst, San Diego, CA, 1992, pp [18] V Kreinovich, H T Nguyen, Y Yam, Fuzzy systems are universal approximators for a smooth function its derivatives, Int J Intell Syst, vol 15, pp , 2000 [19] M Lajo, M J Rio, R Pérez, A note on smooth spproximation capabilities of fuzzy systems, IEEE Trans Fuzzy Syst, vol 9, pp , Apr 2001 [20] C C Lee, Fuzzy logic in control systems: fuzzy logic control part I, IEEE Trans Syst, Man, Cybern, vol 20, pp , Mar/Apr 1990 [21] Z-H Mao, Y-D Li, X-F Zhang, Approximation capability of fuzzy systems using translations dilations of one fixed function as membership function, IEEE Trans Fuzzy Syst, vol 5, pp , Aug 1997 [22] W Pedrycz, Why triangular membership functions, Fuzzy Sets Syst, no 64, pp 21 30, 1994

9 168 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART A: SYSTEMS AND HUMANS, VOL 33, NO 2, MARCH 2003 [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 , 1988 [25] H Varian, Microeconomic Analysis New York: Norton, 1978 [26] H D Vinod 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, P S Tseng, Fuzzy B-spline membership function its applications in fuzzy-neural control, IEEE Trans Syst Man, Cybern, vol 25, pp , 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 J M Mendel, Fuzzy basis functions, universal approximations orthogonal least squares learning, IEEE Trans Neural Networks, vol 3, pp , 1992 [30] P-Z Wang, S Tan, F Song, P Liang, Constructive theory of fuzzy systems, Fuzzy Sets Syst, no 88, pp , 1997 [31] H Ying, Constructing nonlinear variable gain controllers via the Takagi-Sugeno Fuzzy control, IEEE Trans Fuzzy Syst, vol 6, pp , May 1998 [32], General SISO Takagi-Sugeno fuzzy systems with linear rule consequent are universal approximators, IEEE Trans Fuzzy Syst, vol 6, pp , May 1998 [33] X-J Zeng M G Singh, Approximation theory of fuzzy systems SISO case, IEEE Trans Fuzzy Syst, vol 2, pp , May 1994 [34], Approximation theory of fuzzy systems MIMO case, IEEE Trans Fuzzy Syst, vol 3, pp , May 1995 Fakhreddine Karray (SM 99) received the Ing Diplome degree in electrical engineering from University of Tunis (ENIT), Tunis, Tunisia in in 1984 the PhD degree from the University of Illinois, Urbana-Champaign, in 1989 in the area of systems controls He was in the faculty of engineering at the University of British Columbia 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 the Department of Electrical Computer Engineering the Associate Director of the UW s Pattern Analysis Machine Intelligence Laboratory His areas of expertise span the fields of intelligent systems design advanced controls using tools of computational intelligence with application to a wide range of industries in the manufacturing, automotive, telecommunication sectors He has published in these areas refereed articles that appeared in journals, textbooks, encyclopedias conference proceedings He is also the author of seven provisional patents He has consulted advised a number of high tech companies in Canada the USA is the cofounder of Intelligent Mechatronics Systems, Inc VESTEC, Inc Dr Karray has served on several occasions as a member of international program committees of international conferences symposia was the Program Chair of the 2002 IEEE International Symposium on Intelligent Control He is also an Associate Editor Member of the Editorial Board of three technical journals IEEE conference proceedings He is the recipient of three best papers awards, 1998, 1995, 1993, an IEEE Technical Speaker award, Mexico, 1999 the Ontario Premier Research Excellence Award in 2000 He is the Chair of the IEEE Control Systems Society Kitchener Waterloo Chapter 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 BEE degree in 1990, the PhD HDR degrees both in electrical engineering, in , respectively He is now Associate Professor in electrical 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, industrial processes He focuses his research on intelligent pattern recognition, learning, analysis 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 Systems Journal, Integrated Computer Aided Engineering Journal, Systems Analysis Modeling 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, PRS systems Radhia Hassine was born in Tunisia in 1971 She received the MS degree in mathematics in 1993 from the University of Center, Monastir, Tunisia She is currently pursuing the PhD 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 the comparison of the approximation properties of different families of fuzzy logic Mohamed Selmi was born in Tunisia in 1952 He received the MS degree in the area of mathematical physics Hamiltonian systems in 1979 from the University of Paris, Paris, France the PhD degree in 1992, in the area of groups 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 the deformation theory