A Neuro-fuzzy-sliding Mode Controller Using Nonlinear Sliding Surface Applied to the Coupled Tanks System

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1 International Journal of Automation and Computing 06(1), February 009, 7-80 DOI: /s A Neuro-fuzzy-sliding Mode Controller Using Nonlinear Sliding Surface Applied to the Coupled Tanks System Ahcene Boubakir 1, Fares Boudjema Salim Labiod 3 1 Control Laboratory, Military Polytechnic School, Bordj-EL-Bahri, Algiers 16111, Algeria Process Control Laboratory, National Polytechnic School, Avenue Pasteur, Hassen Badi, BP 18, El-Harrach, Algiers, Algeria 3 Laboratory of Modeling and Studies in Electrical Engineering, Faculty of Engineering, University of Jijel, Jijel, Algeria Abstract: The aim of this paper is to develop a neuro-fuzzy-sliding mode controller (NFSMC) with a nonlinear sliding surface for a coupled tank system. The main purpose is to eliminate the chattering phenomenon and to overcome the problem of the equivalent control computation. A first-order nonlinear sliding surface is presented, on which the developed sliding mode controller (SMC) is based. Mathematical proof for the stability and convergence of the system is presented. In order to reduce the chattering in SMC, a fixed boundary layer around the switch surface is used. Within the boundary layer, where the fuzzy logic control is applied, the chattering phenomenon, which is inherent in a sliding mode control, is avoided by smoothing the switch signal. Outside the boundary, the sliding mode control is applied to drive the system states into the boundary layer. Moreover, to compute the equivalent controller, a feed-forward neural network (NN) is used. The weights of the net are updated such that the corrective control term of the NFSMC goes to zero. Then, this NN also alleviates the chattering phenomenon because a big gain in the corrective control term produces a more serious chattering than a small gain. Experimental studies carried out on a coupled tank system indicate that the proposed approach is good for control applications. Keywords: Sliding mode, fuzzy logic, neural networks, coupled tanks system. 1 Introduction Variable structure systems with a sliding mode were discussed first in literature [1] in the former Soviet Union, and have been widely developed in recent years. Sliding mode control has found several applications in various fields such as power electronics [], power electrical systems [3], and robot manipulators [4]. Sliding mode controller (SMC) design comes up with two parts: the equivalent control and the corrective term. An SMC law is designed such that the representative point s trajectories of the closed-loop system are attracted by the sliding surface, and once on the sliding surface they slide towards the origin. As the sliding surface is hit, the system response is governed by the surface; consequently, the robustness to the uncertainty or disturbance is achieved. However, in applications of practical control, the SMC suffers from two main disadvantages [5]. The first one is the chattering, which is the high frequency oscillations of the controller output. The second one is the difficulty in the calculation of the equivalent control. Therefore, a thorough knowledge of the plant dynamics is required for this purpose. In the literature, several methods of chattering reduction have been reported. The approach in [6] places a boundary layer around the switching surface such that the relay control is replaced by a saturation function. Another method in [7] replaces a max-min-type control by a unit vector function. These approaches, however, provide no guarantee of convergence to the sliding mode and involve a tradeoff between chattering and robustness. Reduced chatter- Manuscript received January 11, 008; revised September 16, 008 *Corresponding author. address: ah boubakir@yahoo.fr ing may be achieved without sacrificing robust performance by combining the attractive features of fuzzy control with SMC [8 10]. Fuzzy logic, first proposed by Zadeh [11], has been proven to be a potent tool for controlling ill-defined or parameter-variant plants. Generally, in the automatic control field, fuzzy systems are employed to approximate the plant s unknown nonlinear functions [1,13], or to improve the performances of the controller. In [9], a hybrid controller which consists of a parallel connected SMC and a neuro-fuzzy controller was proposed. The aim of that study was to obtain a controller that eliminates the chattering phenomenon and provides a fast and smooth dynamic response. In [10], a fuzzy logic system was used as adaptation mechanism of the SMC with a proposed chattering index. The idea of the work was to use an adaptative gain in the corrective control term. In the reaching phase, we use big gain and small gain in the sliding mode phase. However, the control law computation is difficult in the case of high order systems. On the other hand, one way of avoiding the computational burden involved in the calculation of the equivalent control is the use of an estimation technique as suggested in [14]. In recent years, much attention has been paid to neural-network-based controllers. The nonlinear mapping and learning properties of neural networks (NNs) are key factors for their use in the control field. In general, an NN controller with the learning rule based on sliding mode algorithm, is used to assure calculation of the equivalent control in the presence of plant uncertainties [5, 15, 16]. This paper proposes an approach of cooperative control based on the combination of fuzzy logic, NNs, and the methodology of sliding mode control. At first, we develop a

2 A. Boubakir et al. / A Neuro-fuzzy-sliding Mode Controller Using Nonlinear Sliding Surface 73 nonlinear sliding surface, and we search the properties that must be fulfilled in order to achieve our control objective. Then, we design the control law to make the developed surface globally attractive and invariant. In order to reduce the chattering phenomenon, a fuzzy logic controller is used to approximate the corrective control term. In the procedure, a fixed boundary layer is adopted and fuzzy logic control is applied within the boundary. Outside the boundary, the sliding mode control is applied to drive the system states into the boundary layer. By tuning the middle membership function width in the fuzzy logic, the system tracking error can converge to a smaller neighborhood of zero than the conventional sliding mode control with boundary layer. To compute the equivalent control, a feed-forward NN is used. The weights of the net are updated such that the corrective control term of the sliding mode goes to zero. Finally, experimental results are given to show the effectiveness and feasibility of the proposed control strategy. System modeling We use the CE105 coupled tanks that consists of two separate vertical tanks (see Fig. 1). Both tanks are interconnected by a flow channel where a rotary valve will be used to vary the sectional area of the channel. Hence, this will change the flow characteristics between the tanks. The first tank is fed by liquid through a DC-motor controlled pump, where the equation relating to the entry voltage u and input flow q 1 is as follows: q 1 = Kp u (1) where K p is the pump gain. The second tank also has an outflow channel whose sectional area can be also modified. Furthermore, the equation that relates to the output voltage signal y, indicated by the level sensor, and the liquid level h, for each tank, can be given as follows where Ks is the sensor gain. y i = Ks h i () the tank sectional area. q 0 = S a 0 gh q 1 = S 1a 1 g(h1 h ) where S 1 and S are the sectional areas of channel 1 and channel, g the acceleration caused by gravity, h 1 the liquid level in tank 1, h the liquid level in tank, a 1 and a 0 are the discharge coefficients of valve 1 and valve, respectively. Some parameters (both nominal and experimental) are shown in Table 1. Table 1 Parameters of the coupled tank system Quantity Symbols Values Tank sectional area As m Channel sectional area S 1 max m Channel sectional area S 1 max m Discharge coefficient a Discharge coefficient a 0 1 Maximal liquid level h max 0.5 m Maximal entry voltage u max 10 V Pump gain Kp 7.5 m 3 /s V Sensor gain Ks 40 V/m Gravity constant g 9.8 m/s Hence, the model of the coupled tanks system can be written as dh 1 = 1 ( ) S 1 a 1 g(h1 h ) + Kp u dt As dh = 1 ) (S 1 a 1 g(h1 h ) S a 0 gh dt As y = Ks h (5) where u = u max, u, 0, if u u max if 0 < u < u max if u 0. The model in (5) can be written in the state-space form as ẋ 1 = f 1(x) ẋ = f (x) + Ka u (7) y = Ks x 1 (4) (6) where x = [x 1, x ] T = [h, h 1] T, and Fig. 1 Layout of the coupled tanks system By using the flow balance equation as indicated in [17], the equations that describe this system are ḣ 1 = 1 ( q1 + q1) As ḣ = 1 (3) As (q1 q0) where q 1 is the input flow rate, q 1 the liquid flow rate between the two tanks, q 0 the outflow liquid rate, andas f 1(x) = β 1 x x 1 β x1 f (x) = β 1 x x 1. The coefficients β 1, β, and Ka are given by Ka = Kp As β 1 = (S1a1 g) As β = (Sa0 g). As (8)

3 74 International Journal of Automation and Computing 06(1), February Design of SMC In this section, the first goal in our topic is to characterize a class of manifold on which the control objective is achieved. We recall that the sliding mode control objective consists of first designing a suitable manifold Ψ(x, t) R m defined by Ψ = {x R n /S(x) = 0} to restrict the state trajectories of the plant to this manifold to result in the desired behavior such as tracking, regulation, and stability; then, determining a switching control law, u(x, t) which is able to drive the state trajectory to this manifold and maintain it on this manifold for all the time. That is, u(x, t) is determined such that the selected manifold Ψ(x, t) is made attractive and invariant. Generally, in sliding mode control, the switching surfaces are linear functions. Slotine and Li [18] gave a form of this sliding manifold which was a Hurwitz polynomial of the error and its derivatives were up to ˆr 1, where ˆr is the relative degree of the output. For the SMC using linear elements, the linear switching surfaces often yield a satisfactory system response, in terms of stability and robustness, to the parameter variations and disturbances. However, with linear switching surface, the speed of transient response is relatively slow [19]. In this paper, the design and analysis of SMC with nonlinear switching surfaces are considered. From the fact that the output y = Ks x 1 is of relative degree two, and in order to obtain static feedback, we define the manifold Ψ(e) as follows: Ψ(e) = {e R S(e) = ė + Λ(e) = 0} (9) with e = y y d, y d = Ks h d, is the tracking error, Λ( ) is any given class C 1 function whose property will be derived below. One has the following result. Proposition 1. Consider the manifold Ψ defined in (9), and assume that Λ( ) is a continuous function such that e Λ(e) > 0, e 0. Then, on the manifold Ψ, the output error e converges at least asymptotically to zero. Proof. Due to the form of manifold Ψ, we have ė = Λ(e). (10) Let us use the Lyapunov function given by V = 0.5e. Its derivative is then V = e Λ(e). (11) In order to make V negative definite, it is enough that e Λ(e) > 0, e 0. Hence, the output error e is bounded and moreover, it tends at least asymptotically to zero. As an example, for function Λ( ), it can be taken as the sigmoid function whose definition is as follows [0] : For a given ε, 0 < ε < 1, a continuous function Λ ε of R R is known as an ε-sigmoid if it obeys the following relations, for z R: 1) Λ ε(z) z > 0, z 0 (1) ) Λ ε(0) = 0 (13) z ε Λ 3) ε(z) z (1 ε) ε (14) z ε 1 Λ ε(z)sgn(z) ε. Hence, Ψ is a suitable manifold for our control system, since the control objective is achieved on it. Let us now design the control law u that makes Ψ attractive and invariant. Proposition. Consider the manifold Ψ defined in (9) and let the control signal u be given by and u = u eq + u c (15) u eq = A 1 0 (x) [B 0(x) + C 0(x)] (16) u c = A 1 0 (x) ( B + δ B0(x) + C 0(x) + m ) sgn (S) (1 δ) (17) where m > 0, B and δ are selected such that B + C B, A δ < 1, with A, B, and C represent, in this case, the parametric uncertainties and unstructured uncertainties: A(x) = A 0(x)(1 + A(x)) with B 0(x) = Ks B(x) = B 0(x) + B(x) C(x) = C 0(x) + C(x) [ ] f (x) f 1(x) β 1 f 1(x) β x x 1 x 1 [ 1 Λ(e) ] ÿ d C 0(x) = (Ks f 1(x) ẏ d ) µ Ks β1 Ka A 0(x) = x x 1 (18) Λ(e) = 1 + e 1 µ e dλ(e) = µ [ 1 Λ(e) ], µ > 0 (19) dt where f i(x), for i = 1,, are given in (8), and the function Λ( ) is given in (19) and characterized in Proposition 1. Then, Ψ is globally attractive and invariant. Proof. Let us consider the following Lyapunov function V = 0.5S, whose time derivative is V = S Ṡ, with Ṡ = B 0 + B + C 0 + C + A 0(x)(1 + A(x)) u. (0) Using the control laws given in (15), V can be written as V = S ( B(x) + C(x)) + S A 0(x)u c+ S A 0(x) A(x)u c + SA 0(x) A(x)u eq S B + A 0(x)S u c + A 0(x) δ Su c + δ A 0(x) S u eq. (1) With the control laws given in (16) and (17), we have S u c = S u c, δ A 0(x) S u eq = δ S B 0(x) + C 0(x). Then, V S B + A 0(x) (1 δ)su c + δ S B 0(x) + C 0(x) m S. () In order to make V negative definite ( V < 0), S 0, it is sufficient to take m > 0. This condition makes S = 0, and hence Ψ is globally attractive. Moreover, if Ṡ = 0, Ψ is invariant.

4 A. Boubakir et al. / A Neuro-fuzzy-sliding Mode Controller Using Nonlinear Sliding Surface 75 with The corrective control term (17) can be given as K > max x u c = K sgn(s) (3) ( A 1 0 (x) ( B + δ B0(x) + C 0(x) + m )). (4) (1 δ) 4 Design of NFSMC 4.1 Structure of the proposed controller In the proposed structure, the corrective term in the sliding controller is approximated by a continuous fuzzy logic control and the equivalent control term, in sliding mode, is computed by an NN. The output of the NN is summed with the corrective term to form the control signal. The corrective control is accepted as a measure of the error to update the weights of the NN [5, 1]. The aim of the learning process of the NN is to minimize the corrective control. This is because in sliding mode the equivalent control is enough to keep the system on the sliding surface and the corrective term is necessary to compensate the deviations from the surface [5, 1] The overall system with the proposed controller is given in Fig.. where T ( S) is the term set of S, and NB, NM, ZR, PM, and PB are labels of fuzzy sets, which are negative big, negative medium, zero, positive medium, and positive big, respectively. For the control output u fuzzy, its term set and labels of the fuzzy sets are defined similarly by T (ũ c) = {NB, NM, ZR, PM, PB} = { F 1 u,, F 5 u}. (7) The membership functions of these fuzzy sets are depicted in Fig. 3. In Fig. 3 (a), r (0, 1] is a coefficient to be used to adjust the input center point, and Φ is the defined boundary layer around the switch surface. (a) Fuzzy partition of the universe of discourse of S (b) Fuzzy partition of the universe of discourse of u fuzzy Fig. 3 Representation of term sets, T ( S) and T (ũ c) From these two term sets, we can build the following fuzzy rules [8] : Fig. The structure of NFSMC 4. Computation of the corrective control The controller in (15) results in high frequency oscillations in its outputs, causing a problem known as chattering. Chattering is undesirable because it can excite the high frequency dynamics of the system. To eliminate chattering, a continuous fuzzy logic control u fuzzy is used to approximate u c. The design of the fuzzy controller begins with extending the crisp sliding surface S = 0 to the fuzzy sliding surface defined by linguistic expression [] : S is ZERO (5) where S is the linguistic variable for S and ZERO is one of its fuzzy sets. In order to partition the universe of discourse of S, the following fuzzy sets are introduced. T ( S) = {NB, NM, ZR, PM, PB} = { FS, 1, FS 5 } (6) R 1 : If S is NB then u fuzzy is PB R : If S is NM then u fuzzy is PM R 3 : If S is ZR then u fuzzy is ZR R 4 : If S is PM then u fuzzy is NM R 5 : If S is PB then u fuzzy is NB. (8) Once the membership functions and fuzzy rules are determined, the final step is the defuzzification, which is the procedure to determine a crisp control for u fuzzy. There are many defuzzification strategies such as the maximum criterion, the mean of maximum, the centre of area, and the weighted average method [8, ]. We use the weighted average method to get the crisp control for u fuzzy. Then u fuzzy = 5 i=1 C f i µ i(s) 5 i=1 µi(s) (9) where C fi is the associated singleton membership function of u fuzzy. Finally, the result of the inference for every S can be written as follows: u fuzzy = K sig S Φ (30)

5 76 International Journal of Automation and Computing 06(1), February 009 where sig(z) = 1 z < 1 z + r 1 1 z < r r z r r z < 0 z 0 z < r r z + 1 r r r z < 1 1 z 1. (31) In Fig. 4, we show the results of the influence of the fuzzy rules with different r values. From Fig. 4, we see that the value of r plays an important role in the shape of this function, and if r = 1 then this function is a saturation function. Fig. 4 Results of the influence of the fuzzy rules 4.3 Computation of the equivalent control The NN is chosen to be a three-layer feed-forward NN, which has one input layer, one output layer and one hidden layer. The structure of inputs and the output of the network are established by the equivalent control equation [16]. The structure of NN used to generate u eq is presented in Fig. 5. From Fig. 5, it is found that the equivalent control is computed by using the iterative gradient algorithm to minimize the mean square error between the desired and actual states. The symbols used in Fig. 5 are defined as follows. Let Z i be the input to the i-th node in the input layer, Y net j be the input to the j-th node in the hidden layer, and the output of the hidden layer be Y out j. Similarly, the input and output of the output layer are designated as Unet and Uout, respectively. Furthermore, W z i,j means the weight between the input layer and the hidden layer, W yj means the weight between the hidden layer and the output layer. The values can be computed as Y net j = N W z i,j Z i (3) i=1 Y out j = g (Y net j) (33) Unet = M W y j Y out j (34) j=1 Uout = g (Unet) (35) û eq(t) = K eq Uout (36) g(x) = 1. (37) 1 + e x where the number of N represents the total number of the input neurons, and M represents the total number of the hidden neurons. In practice, the number of the hidden neurons can be selected as two times the number of neurons in the input layer. The activation function g( ) is selected as a sigmoid transfer function, as defined in (37). K eq is a constant that represents the maximum available value of the equivalent control. Thus, û eq is the estimated value of the equivalent control. In order to prevent the equivalent control from exceeding the maximum bound of the actuator, or reaching an unreasonably large value, the output of the neural network is kept in [ 1, 1]. In a general NN, the backpropagation uses the gradient descent method to establish the multilayer feedforward network. The training processes use iterative gradient algorithms to minimize the mean square error between the actual output and the desired output, i.e., to minimize the cost function selected as the difference between the desired and the estimated equivalent controls. Hence, a simple cost function is defined as follows Fig. 5 The structure of NN to estimate the equivalent control E = 1 [ueq (t) ûeq (t)]. (38) The objective is to minimize the error function E by taking the error gradient with respect to the weights. The weights are then updated by using W y j(t) = W y j(t 1) α W z i,j(t) = W z i,j(t 1) α W y j (39) W z i,j (40) where α is a constant that denotes the learning rate parameter of the backpropagation algorithm. Moreover, the two terms / W y j and / W z i,j can be derived as follows:

6 A. Boubakir et al. / A Neuro-fuzzy-sliding Mode Controller Using Nonlinear Sliding Surface 77 1 = (u eq û eq) K eq W yj ( 1 Uout ) Y out j (41) 1 ( = (u eq û eq) K eq 1 Uout ) W y j W z i,j 1 ( 1 Y out ) j Zi. (4) Notice that the actual equivalent control ueq(t)in (41) and (4) is unknown. Hence, (39) and (40) cannot be calculated. In order to overcome this problem, the value of corrective control u fuzzy is utilized to replace ueq ûeq. The reason is that the characteristics of ueq ûeq and corrective control are similar [5, 16]. 1 = u fuzzy K eq W y j ( 1 Uout ) Y out j (43) 1 ( =1 u fuzzy K eq 1 Uout ) 1 ( W y j 1 Y out ) j Zi. W z i,j (44) 5 Experimental studies In order to verify the effectiveness and the efficiency of the proposed NFSMC, an application to the coupled tank system has been conducted. The reference level h d used to evaluate the response of the NFSMC applied to the coupled tank system was a pulse train whose amplitude varied between.5 cm and 7.5 cm and changed every T/ = 00 s with duty cycle τ = 0.5. The plant and controller parameters used for the experimental are given in Tables 1 and, respectively. The parameter S 1 was kept at 100%, channel 1 was completely open, and the parameter S was varying (see Fig. 6). In practice, the sectional area of the outflow channel is manually changed with the rotary valve. Fig. 7 shows the experimental test bench. The control law was implemented on a PC Pentium II at 00 MHz, equipped with a dspace DS110 controller board, using Matlab and Simulink with sampling time 0. s. The sliding manifold that represents a desired system dynamics was given by (9) and (19). In the case of the classical SMC, the corrective control is given by (3), and equivalent control is given by (16) and (18). In the proposed NFSMC, the corrective control and equivalent control are computed by (9) and (36), respectively. For practical reasons, two Chebyshev low-pass filters were used to reduce the amount of noise caused by the oscillation of the liquid in the tanks provoked by some internal liquid flows and turbulence. The first filter received the output signal of tank 1, and the second filter received the output signal of tank. For each filter, the design parameters were a cut-off frequency of 0.5 Hz with a cut-off gain of 0 db. Table Parameters of the controller and sliding manifold used in experiment Application Symbols Values Sliding surface µ (Gradient of sigmoid function) 0.5 Fuzzy controller Neural network Φ(Boundary layer) 0.1 r (Coefficient used to adjust the centre point) 0.8 K (Feedback gain) 8 N(Number of inputs) 3 M (Number of hidden neurons) 6 Number of outputs 1 K eq (The maximum available value of the equivalent control) α (Learning rate) Fig. 6 Section area of the outflow channel The input of the neural network (designated as Z) consisted of the error and actual states, as Z = [h 1 h e] and all the network weights were initialized to small random values between [ 0.05, 0.05]. Moreover, in order to prevent from blowing the weights up, the value of u fuzzy which was used to adapt the weights, was divided by u max, which is the maximum available value of the control. Fig. 7 Experimental test bench The results of using the conventional sliding mode controller are shown in Fig. 8 and the results obtained using the proposed control strategy is shown in Fig. 9. From the experimental results, we can find that the control result of

7 78 International Journal of Automation and Computing 06(1), February 009 (a) (a) (b) (b) (c) (c) (d) (d) (e) (e) Fig. 8 Experimental results of SMC applied to the coupled tank system Fig. 9 Experimental results of NFSMC applied to the coupled tank system

8 A. Boubakir et al. / A Neuro-fuzzy-sliding Mode Controller Using Nonlinear Sliding Surface 79 the conventional SMC produces a serious chattering phenomenon, as in Figs. 8 (a), (b), and (c). On the contrary, the chattering phenomenon of the controlled system was suppressed in the proposed controller, as shown in Figs. 9 (a), (b), and (c). Moreover, in the NFSMC, we did not need to compute the dynamical equation of the system and the equivalent control was estimated by the NN, as shown in Fig. 9 (d). Furthermore, the proposed controller is a robust controller since the variation of the sectional area of the outflow channel had no influence on the control performances. To compare the performances of NFSMC with SMC, we define two cost functions in Table 3, with p being the number of measured data. The experimental results for each performance index are also given in Table 3. Table 3 Comparative study with p = 4500 Controllers J 1 = 1 p e i J = 1 p u i i=1 i=1 NFSMC SMC By comparing the experimental results, it can be said that the proposed control strategy gives better performances than using the conventional sliding mode controller. 6 Conclusions In this paper, an NFSMC was proposed for a coupled tank system, and experimental results were presented. First, a general class of manifolds for sliding mode control of the coupled tank system was developed. The proprieties of sliding surface, ensuring the control objective, were derived. Second, the SMC, using the proposed nonlinear sliding surface, was designed by selecting a Lyapunov function. The design yielded an equivalent control term plus an addition control term. Third, it was surveyed how the fuzzy controller was used to compute the corrective control and how the NN was used to compute the equivalent control. To eliminate chattering, a continuous fuzzy logic control was used to approximate the corrective control. The structure of the NN that estimates the equivalent control was a standard three layer-feed-forward NN with the backpropagation adaptation algorithm. The corrective control was accepted as a measure of error to update the weights of the NN. The proposed method has the following advantages: 1) There is no need to know the dynamical equation of a system to compute the equivalent control. ) Chattering and the excessive activity of the control signal are eliminated without a degradation of the tracking performance. 3) The learning process is online. Learning and calculation of the equivalent control signal are carried out simultaneously. The experimental results presented in this paper indicate that the suggested approach has considerable advantages compared to the classical sliding mode control. These characteristics make it a promising approach for control applications. 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9 80 International Journal of Automation and Computing 06(1), February 009 [15] M. A. Hussain, P. Y. Ho. Adaptive Sliding Mode Control with Neural Network Based Hybrid Models. Journal of Process Control, vol. 14, no., pp , 004. [16] C. H. Tsai, H. Y. Chung, F. M. Yu. Neuro-sliding Mode Control with Its Applications to Seesaw Systems. IEEE Transactions on Neural Networks, vol. 15, no. 1, pp , 004. [17] P. Wellstead. TecQuipment CE105 Coupled Tanks Apparatus, Control Systems Centre, Manchester, UK, [18] J. J. Slotine, W. Li. Applied Nonlinear Control, Prentice Hall, [19] D. S. Lee, M. G. Kim, H. K. Kim, M. J. Youn. Controller Design of Multivariable Variable Structure Systems with Nonlinear Switching Surfaces. IEE Proceedings: Control Theory and Applications, vol. 138, no. 5, pp , [0] N. Yeganefar, M. Dambrine, A. Kokosy. Stabilisation pratique par modes glissants pour un système linéaire à retard. In Proceedings of Conférence Internationale Francophone D Automatique, Tunisia, CD-ROM, 004. (in French) [1] M. Ertugrul, O. Kaynak. Neural Computation of the Equivalent Control in Sliding Mode for Robot Trajectory Control Applications. In Proceedings of IEEE International Conference on Robotics and Automation, Belgium, vol. 3, pp , [] J. Z. Liu, W. J. Zhao, L. J. Zhang. Design of Sliding Mode Controller Based on Fuzzy Logic. In Proceedings of the 3rd IEEE Conference on Machine Learning and Cybernetics, IEEE Press, Shanghai, PRC, pp , 004. logic. Ahcene Boubakir received the B. Eng. degree in automatic control from University of Jijel, Jijel, Algeria, in 004, M. Sc. degree in automatic control and robotics from Polytechnic School (EMP), Algiers, Algeria, in 007. He is currently a Ph. D. candidate in automatic control at the National Polytechnic School. His research interests include nonlinear control, sliding mode control, and fuzzy Fares Boudjema received the M. Eng. degree in electrical engineering from the Ecole Nationale Polytechnique, Algiers, Algeria, in 1985, the DEA degree, and the Ph. D. degree in automatic control from the Université Paul Sabatier, Toulouse, France, in 1987 and 1991, respectively. In 1991, he joined the Department of Electrical Engineering of the Ecole Nationale Polytechnique, Algiers, as an assistant professor. He was promoted to associate professor in 1994, and professor in 000. His research interests include application of sliding mode control, artificial neural network control, fuzzy control, and decentralized control in the field of the electrical machines, power systems, and robotics. Salim Labiod received the B. Eng. M. Sc. and Ph. D. degrees in control engineering from National Polytechnic School of Algiers (ENP), Algeria, in 1995, 1998 and 005, respectively. Since 1998, he has held teaching and research positions in the Faculty of Engineering at the University of Jijel, Algeria, where he is currently an associate professor. His research interests include nonlinear control, adaptive control, and fuzzy control.