IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 15, NO. 3, MAY 2004 693 Robust and Adaptive Backstepping Control for Nonlinear Systems Using RBF Neural Networks Yahui Li, Sheng Qiang, Xianyi Zhuang, and Okyay Kaynak, Fellow, IEEE Abstract—In this paper, two different backstepping neural network (NN) control approaches are presented for a class of affine nonlinear systems in the strict-feedback form with unknown nonlinearities. By a special design scheme, the controller singularity problem is avoided perfectly in both approaches. Furthermore, the closed loop signals are guaranteed to be semiglobally uniformly ultimately bounded and the outputs of the system are proved to converge to a small neighborhood of the desired trajectory. The control performances of the closed-loop systems can be shaped as desired by suitably choosing the design parameters. Simulation results obtained demonstrate the effectiveness of the approaches proposed. The differences observed between the inputs of the two controllers are analyzed briefly. Index Terms—Adaptive control, backstepping, neural network (NN), robust adaptive control, uncertain strict-feedback system. I. INTRODUCTION N recent adaptive and robust control literature, numerous approaches have been proposed for the design of nonlinear control systems. Among these, adaptive backstepping constitutes a major design methodology [1]–[3]. The idea behind backstepping design is that some appropriate functions of state variables are selected recursively as pseudocontrol inputs for lower dimension subsystems of the overall system. Each backstepping stage results in a new pseudocontrol design, expressed in terms of the pseudocontrol designs from the preceding design stages. When the procedure terminates, a feedback design for the true control input results, which achieves the original design objective by virtue of a final Lyapunov function, formed by summing the Lyapunov functions associated with each individual design stage [1]. The backstepping design provides a systematic framework for the design of tracking and regulation strategies, suitable for a large class of state feedback linearizable nonlinear systems. Integrator backstepping is used to systematically design controllers for systems with known nonlinearities with mismatched conditions [2]. The approach can be extended to handle systems with unknown parameters via adaptive backstepping [2], [4], [5]. Apart from the systematic approach used, another important feature of it is its ability to shape performance [1], [5]. I Manuscript received September 10, 2002; revised December 23, 2003. This work was supported by the Youth Scientific Foundation of Heilongjiang Province, China, by Project QC02C40. Y. Li, S. Qiang, and X. Zhuang are with Department of Control Science and Engineering, Harbin Institute of Technology, Harbin 150001, China (e-mail: lyahui2002@yahoo.com.cn; zhx@hit.edu.cn). O. Kaynak is with Department of Electrical and Electronic Engineering, Bogazici University, 80815 Bebek-Istanbul, Turkey (e-mail: kaynak@boun.edu.tr). Digital Object Identifier 10.1109/TNN.2004.826215 However, in spite of the merits explained, there are some problems in the backstepping design method. A major constraint is that certain functions must be “linear in the unknown parameters” [6], [7], which may not be satisfied in practise. Furthermore, some very tedious analysis is needed to determine “regression matrices”[6]. In the case of backstepping adaptive control, the problem of determining and computing the regression matrices becomes even more acute. In [8], for example, for the relatively simple application to dc motor control, one will notice that the regression matrix almost covers one full page in the IEEE TRANSACTIONS ON CONTROL SYSTEM TECHNOLOGY. The very rapid developments described in adaptive and robust control techniques are accompanied by an increasing in the use of neural networks (NNs) for system identification [9], [10] or identification-based control [11]–[13]. With the help of neural networks, the linearity-in-the-parameter assumption of nonlinear function and the determination of regression matrices can be avoided. It is due to this that in the last few years, a large number of backstepping design schemes are reported that combine the backstepping technique with adaptive NNs [6], [7], [14]–[20]. Although significant progress has been made by combining backstepping methodology with NN technologies, there are still some problems that need to be solved for practical implementations. For example. in almost all the approaches reported in the literature, in order to avoid the controller singularity problem it is assumed that the gain functions (see (1) in Section II) are constants [6], [7], [14]–[17] or known functions [6]. However, this assumption cannot be satisfied in many cases. In some work, for example in [18], the authors assume the gain functions to be unknown and adopt NN structures to approximate them. In order to avoid the possible divergence of the weights of the neural networks during on-line tuning, the discontinuous projections with fictitious bounds have to be used in design. In [19], gain functions are assumed to be unknown and a backstepping design is proposed that incorporates adaptive neural network techniques. However, due to the integral-type Lyapunov function introduced, this approach is complicated and difficult to use in practice. In [20], Ge et al. propose another simpler scheme under the same conditions. Nevertheless, because the derivatives of the virtual controllers are included in NNs, the NNs are difficult to realize and calculate. Variable structure control is one of the main methods used in the literature to overcome the uncertainty of systems. In backstepping design, this approach is difficult to use, because the derivatives of virtual controllers are all included in the actual controller. Therefore, robustness issues are rarely addressed in 1045-9227/04$20.00 © 2004 IEEE Authorized licensed use limited to: IEEE Xplore. Downloaded on February 19, 2009 at 04:51 from IEEE Xplore. Restrictions apply. 694 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 15, NO. 3, MAY 2004 backstepping design. In this paper, an alternative, simpler design is proposed, which can completely avoid the singularity problem. Based on this, a class of continuous functions is introduced to ensure robustness of in backstepping design. The paper is organized as follows. In Section II, a description of the system is given and radial basis function (RBF) neural networks are briefly explained. Adaptive backstepping design is described in Section III and simulated on a nonlinear system. Section IV proposes an approach to ensure a degree of robustness in the controller and the effectiveness of the approach is demonstrated by the simulation studies on the same system as used in Section III. Finally, conclusions are given in Section V. II. PROBLEM FORMULATION ( is an appropriate integer) where is a compact subset of , there exists a RBF vector and a weight and such that , . vector The quantity is called the network . reconstruction error and obviously The optimal weight vector defined above is a quantity only for analytical purposes. Typically is chosen as the value of that minimizes over , that is (2) The Gaussian functions are employed as basis functions, in the same form as in [20], which are located on a regular grid that contains the subset of interest of the state space. A. System Description III. ADAPTIVE NN CONTROL The model of many practical nonlinear systems can be expressed in or transformed into a special state-space form (1) where , , , are state variables, system input and output, respectively. The control objective is to design an adaptive NN controller for system (1) such that 1) all the signals in the closed-loop remain semiglobally uniformly ultimately bounded and 2) the output y follows a desired trajectory , which and whose derivatives up th order are bounded. to the Note that in the following derivation of the adaptive neural controller, NN approximation is only guaranteed with some compact sets. Accordingly, the stability results obtained in this work are semiglobal in the sense that, as long as desired, there exists controllers with sufficient large number of NN nodes such that all the signals in the closed-loop remain bounded. , are smooth functions, they are thereSince fore bounded within some compact set. Accordingly, we can make the following two assumptions as commonly being done in the literature. are bounded, i.e., there exist Assumption 1: The signs of such that , , constants . The above assumption implies that the smooth functions are strictly either positive or negative. Without losing generality, , . we shall assume, Assumption 2: There exist constants such that, , . A. Controller Design The detailed design procedure is described in the following steps. For clarity and conciseness, Steps 1 and 2 are described with detailed explanations, while Step and Step are simplified, with the relevant equations and the explanations being omitted. and define . Its derivative Step 1: Let is (3) by viewing as a virtual control input. Equation (3) can be transformed into the following form: (4) as follows: Let us choose controller (5) is constant. Substituting (5) into (4), is obtained. So, there exists a Lyapunov function , such that . Therefore, is asymptotically stable. However, since the functions and are unknown, the desired controller cannot be implemented in practice. Instead, a NN-based virtual controller can be used as follows: where (6) where mate Defining and and , are RBF NNs used to approxi, respectively. can be obtained as B. Function Approximation Using RBF Neural Networks The control design presented in this paper employs RBF NNs to approximate the nonlinear functions in system (1). They are , where is a vector of the general form of adjustable weights and a vector of RBF’s. Their ability to uniformly approximate smooth functions over compact sets is well documented in the literature (for example [21]). In general, , it has been shown that given a smooth function (7) and are the optimal weight vectors of where . The neural reconstruction error Authorized licensed use limited to: IEEE Xplore. Downloaded on February 19, 2009 at 04:51 from IEEE Xplore. Restrictions apply. and LI et al.: ROBUST AND ADAPTIVE BACKSTEPPING CONTROL is bounded, i.e., there exists a constant such that . Throughout the paper, we introduce and neural networks and define their reconstruction errors as 695 as (15) where . Like in the case of . Substituting (6) into (7), we get , is bounded, i.e., Because choosing such that the following inequality: , by , we have (8) , . Through out this paper, we where shall define . Consider the following Lyapunov candidate: (16) (9) where the coupling term will be canceled in the next step. Step 2: This step is to make the error between and as small as possible. Differentiating gives where matrices. The derivative of , are adaptive gain is (17) Similarly, taking the virtual controller as of the form (18) and substituting it into (17), we will have (10) (19) Consider the Lyapunov function candidate Consider the following adaptation laws: (20) (11) where , are small constants. Formulas (11) are so-called -modification, introduced to improve the robustness in the presence of the approximation error and avoid the weight parameters to drift to very large values. Let , with and . Then, (10) becomes where matrices. The derivative of , are adaptive gain is (12) By completion of squares, we have (21) where . Consider the following adaptation laws: (13) (22) (14) where and are small constants. Let , where and . Substituting (16) and (22) into (21), and with some completion of squares and straightforward Authorized licensed use limited to: IEEE Xplore. Downloaded on February 19, 2009 at 04:51 from IEEE Xplore. Restrictions apply. 696 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 15, NO. 3, MAY 2004 derivation similar to those employed in Step 1, the derivative of is obtained as Similarly, letting (31) and substituting it into (30), gives (23) (32) Consider the overall Lyapunov function candidate where is chosen such that (33) And consider the following adaptation law: Step : In a similar fashion, we can design to make the error as a virtual controller small as possible. Differentiating gives (34) where (24) . where Similarly, let the virtual controller to be of the form and are small constants, , are gain matrices. Let , where and . By using (29), (32) and (34), and with some completion of squares and straightforward derivation similar to those employed in the former steps, the derivative of becomes (25) Then we have (35) (26) Consider the Lyapunov function candidate where is chosen such that Let choose such that (27) and consider the following adaptation laws: (28) where and are small constants. Let , where and . By using (23), (26), and (28), and with some completion of squares and straightforward derivation similar to those employed in the former steps, the derivative of becomes choose where , , and . If we , i.e., choose such that , where is a positive constant, and such that is the largest eigenvalue of matrices, and . Then, from (35), we have the following inequality: (36) (29) where is chosen such that (37) Step : This is the final step. Differentiating the error , we will have (30) The following theorem shows the stability and control performance of the closed-loop adaptive system. Theorem 1: Consider the closed-loop system consisting of (1) and the known reference signal, the controller (31), and the NN weight updating laws (11), (22), (28) and (34). Assume that , and with there exists sufficiently large compact sets proper dimensions, such that , and Authorized licensed use limited to: IEEE Xplore. Downloaded on February 19, 2009 at 04:51 from IEEE Xplore. Restrictions apply. LI et al.: ROBUST AND ADAPTIVE BACKSTEPPING CONTROL 697 for all . Then for bounded initial conditions, we have the following: 1) all signals in the closed-loop system remain bounded; converges to a 2) the output tracking error small neighborhood around zero by an appropriate choice of the design parameters. Proof: 1) From (36), we have (38) Let . Then, we have (45) that is (46) which implies that given , there exists such that for all , the tracking error satisfies where (47) (39) and is the minimum of , . Therefore, the derivative of global Lyapunov function is negative as long as (40) or (41) or (42) According to a standard Lyapunov theorem extension [22], these demonstrate the uniformly ultimately bound, and edness (UUB) of , and . Since are bounded, we have that is bounded. From , , and the definitions of (6), (18), and (25), we have that virtual controls remain bounded. Using (31), we conclude that control is also bounded. According to assumption of the system and are continuous, the that the function function are bounded in any certainty compact. Thus, the , are bounded, optimal weights and , and the weights and of the NNs are bounded because of (40) and (41). So, all the signals in the closed-loop system remain bounded. , then (37) satisfies 2) Let (43) From (43), we have (44) where is the size of a small residual set which depends on the NNs approximation error and controller parameters , , , and . It is easily seen that the increase in the control gain and adaptive gain , and NN node numbers will result in a better tracking performance. Remark 1: In [20], one NN is adopted to approximate the in every design step. nonlinear function However, because the derivatives of the virtual control are included in the NNs, the dimensions of the input vectors of the NNs become twice as much as those of the corresponding state vectors and these additional inputs must be computed online too. Therefore, the approach is still difficult to implement and apply in practice. In this paper, although two NNs are and adopted to approximate the nonlinear functions , respectively, in every “step”, there are no dimensional increments and no additional parameters must be calculated. Compared with the approach in [20], the method presented in this paper is much simpler to understand and apply in practice. For is two-dimensional (2-D). So, example, we assume that the the input vector of nonlinear function in literature [20] is 3-D at least, while those of nonlinear functions and are still 2-D. Therefore, if we are given five fuzzy sets for every term of the input vectors, the node number will be 125 although just one neural network . is adopted to approximate the function However, each of the neural networks adopted in this paper is node one, although two networks must be adopted to 25 approximate function and , respectively. So, we can draw the conclusion that the controller is simpler to for practical realizations and computational burden is not excessive. Remark 2: In the above analysis, it is clear that the uniform ultimate boundedness of all the signals are guaranteed large enough, such that by choosing . Moreover, it can be seen that 1) increasing might lead to larger , and increasing will reduce , thus, increasing will lead to smaller and 2) decreasing and will help to reduce , both of which will help to reduce the size will lead to a high gain control of . However, increasing scheme. On the other hand, though and is required to be chosen as a small positive constant when applying -modification [23], a very small and may not be enough to prevent the NN weight estimates from drifting to very large values in the presence of NN approximation errors [24], where the large Authorized licensed use limited to: IEEE Xplore. Downloaded on February 19, 2009 at 04:51 from IEEE Xplore. Restrictions apply. 698 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 15, NO. 3, MAY 2004 Fig. 1. The output of system under the adaptive controller. and might result in a variation of a high gain control. Therefore, in practical applications, the design parameters should be adjusted carefully for achieving suitable transient performance and control action. Remark 3: The adaptive NN controller (31) and the adaptation laws (11), (22), (28), and (34) are highly structural, and independent of the complexities of the system nonlinearities. Thus, it can be applied to other similar plants without repeating the complex controller design procedure for different system nonlinearities. In addition, such a property is particularly suitable for parallel processing and hardware implementation in practical applications. B. Simulation Here, a simple simulation is presented to show the effectiveness of the approach proposed above. The model of the system is given as Fig. 2. The trajectory of the adaptive controller. (a) controller values in the interval [0, 20]. (b) controller values in the interval [1, 80]. we can see that good tracking performance is obtained. Fig. 2(a) shows the trajectory of the controller and Fig. 2(b) is magnification of the partition of it when the system is in tracking steady-state phase. (48) where and are states, and is the output of the system, respectively. The initial conditions is and the desired reference signal of the system is . In this paper, all the basis functions of the NNs have the form [20] (49) where field and is the center of the receptive is the width of the Gaussian function. The NNs , and all contain 13 nodes (i.e., ), with centers evenly spaced in [ 6, 6], and widths . NN contains 169 nodes ), with centers evenly spaced in (i.e., , and widths . The design parameters of the above controller are , , . The initial weights and are all given arbitrarily in [ 1, 1], and and in [0, 1]. Figs. 1 and 2 show the simulation results of applying controller (31) to (47) for tracking desired signal . From Fig. 1, IV. ROBUST ADAPTIVE NN CONTROL A. System Description From the proof of Theorem 1 and the result of simulations, we can conclude that there exists a certain amount of error between the output of system and the desired signal, although tracking performance of this system is fairly good. It may also be that the tracking requirements are not met under some conditions. So, in this section, we will present another design approach, which can guarantee much higher tracking accuracy and have the property of “robustness.” , Here, virtual controllers are denoted by the , which satisfies and Authorized licensed use limited to: IEEE Xplore. Downloaded on February 19, 2009 at 04:51 from IEEE Xplore. Restrictions apply. (50) LI et al.: ROBUST AND ADAPTIVE BACKSTEPPING CONTROL 699 Equation (50) can be obtained by a similar design procedure as is inused in Section III, except that a term troduced in every virtual and actual controller for robustness. During the stability analysis of the closed-loop system, for simwith the following plicity, we will replace the function , depicted in Fig. 3 saturation function sat (51) where and are all positive constants, . are continuous, So they can Obviously, virtual controllers be applied in backstepping design. Assumptions 3: The ideal weights of NNs are bounded by known positive values so that Fig. 3. The function tanh(x) and sat(x). and , (55) can be By combining (39), where . Let us suppose that Assumptions 1, 2, and 3 are all satisfied. The following theorem can then be stated. Theorem 2: Take the control input (49) with NN weight tuning be provided by transformed into (56) (52) where and are positive constants. Then the NN weights are , , are semiglobally UUB, and the errors and asymptotic stable. By choosing suitable parameters , , the tracking error of the system can converge into an arbitrarily small compact set. with in (7) and substiProof: In Step 1, replacing tuting (7) and (51) into (10), then choosing in (10) such that , we can obtain where is the minimum of , Applying Schwartz inequality to (56) . (57) we have (53) In Step 2, replacing with in (17) and substituting (17) in (21) such that and (51) into (21), then choosing , we can obtain (58) Obviously, if (59) (54) In a similar fashion, in Step , we can obtain that the overall Lyapunov function candidate, satisfying the following inequality: the derivative of Lyapunov candidate is negative. Thus, the NN’s weights are UUB. From (51), when all the error signals satisfy the derivative of becomes (55) Authorized licensed use limited to: IEEE Xplore. Downloaded on February 19, 2009 at 04:51 from IEEE Xplore. Restrictions apply. (60) 700 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 15, NO. 3, MAY 2004 Fig. 4. The second order derivatives of tanh(kx) (dashed line: k = 1, solid line: k = 2, dash-dot line: k = 3). Fig. 5. The output of system under the robust adaptive controller. So, if (61) is negative. So, at least, the is satisfied, the derivative of errors converge into a compact set (62) Remark 4: In theorem 2, the term “semiglobally” asymptotically stable isused because the NNs can only approximate the nonlinear function in some compact set. If we can find NNs which approximate the nonlinear function in all system state space, the approach can guarantee global asymptotic stability. Remark 5: The procedure of proof indicate that the approach is also effective for some other kinds of uncertainties ) or disturbances , , (for example besides neural reconstruction errors. So, it can be set to be “robust” to some extent. Remark 6: It is seen [from the equation before (60)] that the , are, the smaller will larger the parameters , the tracking errors be. However, for high-order systems, due to the existence of high-order derivatives in the virtual controller, too large values will require a quick response of the controller and the output will take large values when the error vector is in a small compact set (see Fig. 4). This is difficult to realize in practice. Besides, parameters , effect the tracking performance. and should be chosen suitably So, both the parameters during the design stage. Fig. 6. The trajectory of the adaptive controller. (a) Controller values in the interval [0, 20]. (b) controller values in the interval [1, 20]. B. Simulation Here, the plant simulated is again (48) and the objective is to . guarantee that the output tracks the desired signal For the simulation studies, all the parameters and the NNs are kept the same as those in Section III, except for and . Fig. 5 and Fig. 6 show the simulation results. From Fig. 5, we can draw the conclusion that this robust approach can guarantee much higher tracking accuracy than the adaptive one. Comparing Fig. 2 to Fig. 6, it can be seen that: 1) the robust controller has a larger maximum than the adaptive case during the tracking transient-state phase and can make the system to track the desired signal faster due to the terms introduced for robustness; and 2) the robust controller has smaller values when the system is in steady-state phase due to the smaller steady-state errors. V. CONCLUSION In this paper, two backstepping NN control approaches are presented for a class of affine nonlinear systems in strict feedback form with unknown nonlinearities. By a special design Authorized licensed use limited to: IEEE Xplore. Downloaded on February 19, 2009 at 04:51 from IEEE Xplore. Restrictions apply. LI et al.: ROBUST AND ADAPTIVE BACKSTEPPING CONTROL scheme, both of the approaches avoid the controller singularity problem perfectly. The signals in the closed loop in the two methods are all guaranteed to be semiglobally uniformly ultimately bounded and the outputs of the system are both proved to converge to a small neighborhood of the desired trajectory. The control performances of the closed-loop systems can shaped by suitably choosing the design parameters. Simulation results carried out on the same nonlinear system demonstrate the effectiveness of the proposed approaches. The differences observed are analyzed briefly. REFERENCES [1] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design. New York: Wiley, 1995. [2] I. Kanellakopoulos, P. V. Kokotovic, and A. S. Morse, “Systematic design of adaptive controller for feedback linearizable system,” IEEE Trans. Automat. Contr., vol. 36, pp. 1241–1253, 1991. [3] P. V. Kokotovic, “The joy of feedback: nonlinear and adaptive,” IEEE Contr. Syst. Mag., vol. 12, pp. 7–17, 1992. [4] M. Krstic, I. Kanellakopoulos, and P. V. 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Neural Networks, vol. 11, pp. 1178–1187, 2000. [16] J. Y. Choi and J. A. Farrell, “Adaptive observer backstepping control using neural networks,” IEEE Trans. Neural Networks, vol. 12, pp. 1103–1112, 2001. [17] S. Jagannathan, “Control of a class of nonlinear discrete time systems using multilayer neural networks,” IEEE Trans. Neural Networks, vol. 12, pp. 1113–1120, 2001. [18] J. Q. Gong and B.Bin Yao, “Neural network adaptive robust control of nonlinear systems in semi-strict feedback form,” Automatica, vol. 37, pp. 1149–1160, 2001. [19] T. Zhang, S. S. Ge, and C. C. Hang, “Adaptive neural network control for strict-feedback nonlinear systems using backstepping design,” Automatica, vol. 36, pp. 1835–1846, 2000. [20] S. S.Shuzhi S. Ge and C. Wang, “Direct adaptive NN control of a class of nonlinear systems,” IEEE Trans. Neural Networks, vol. 13, 2002. [21] R. M. Sanner and J. J. E. Slotine, “Gussian networks for direct adaptive control,” IEEE Trans. Neural Networks, vol. 3, pp. 837–863, 1992. [22] K. S. Narendra and A. M. Annaswamy, “A new adaptive law for robust adaptation withour persistent excitation,” IEEE Trans. Automat. Contr., vol. 32, pp. 134–145, 1987. 701 [23] P. V. Kokotovic, “Bode lecture: the joy of feedback,” IEEE Contr. Syst. Mag., vol. 3, pp. 7–17, June 1992. [24] G. Cybenko, “Approximation by superpositions of a sigmoidal function,” Math. Contr., Signals, Syst., vol. 2, no. 4, pp. 303–314, 1989. Yahui Li was born in Hebei Province, China, on Feburary 1, 1974. He received the M.S. degree in material science, in 2000 and the Ph.D. degree in control theory and control engineering, in 2003, both from the Harbin Institute of Technology (HIT), China. His research focuses on nonlinear system control, adaptive control, and intelligent control. Sheng Qiang received the M.S. degree in pattern recognition and intelligent control in, 1998, and the Ph.D. degree in control theory and control engineering, in 2004, both from the Harbin Institute of Technology (HIT), China. He is an Associate Professor of Control Science and Engineering Department, HIT. He has published more than 20 papers in journals and international conference proceedings. His research interests include soft computing methods, computer control, and process optimization. Dr. Qiang is a member of the IPC of the 12th Mediterranean Conference on Control and Automation and the ITC of the 2005 IEEE Midsummer Workshop on Soft Computing in Industrial Applications. Xianyi Zhuang was born in Jilin Province, China, in 1936. He received the B.Sc. degree in mathematics science from the University of Northeast Normal, China. He is a Professor with the Department of Control Science and Engineering, Harbin Institute of Technology, China. His research focuses on adaptive control, intelligent control, and the development of high degree accuracy servo systems. Okyay Kaynak (M’80–M’83–SM’90–F’03) received the B.Sc. degree with first class honors and the Ph.D. degree in electronic and electrical engineering from the University of Birmingham, U.K., in 1969 and 1972, respectively. From 1972 to 1979, he held various positions within the industry. In 1979, he joined the Department of Electrical and Electronics Engineering, Bogazici University, Istanbul, Turkey, where he is presently a Full Professor. He served as the Chairman of the Computer Engineering Department for three years, of the Electrical and Electronic Engineering Department for two years, and was the Director of Biomedical Engineering Institute for one year. Currently, he is the UNESCO Chair on Mechatronics and the Director of Mechatronics Research and Application Centre. He has held long-term Visiting Professor/Scholar positions at various institutions in Japan, Germany, the U.S., and Singapore. His current research interests are in the fields of intelligent control and mechatronics. He has authored three books and edited five. He has also authored or coauthored more than 200 papers that have appeared in various journals and conference proceedings. Dr. Kaynak was the President of the IEEE Industrial Electronics Society during 2002–2003. He is now of the Vice Presidents of the Neural Networks Society. He serves as an Associate Editor of both the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS and the IEEE TRANSACTIONS ON NEURAL NETWORKS and additionally he is on the Editorial or Advisory Boards of a number of scholarly journals. Authorized licensed use limited to: IEEE Xplore. Downloaded on February 19, 2009 at 04:51 from IEEE Xplore. Restrictions apply.