International Journal of Computers, Communications & Control Vol. III (2008), No. 1, pp. 6-20 Robust Fuzzy Sliding Mode Controller for Discrete Nonlinear Systems Hafedh Abid, Mohamed Chtourou, Ahmed Toumi Abstract: In this work we are interested to discrete robust fuzzy sliding mode control. The discrete SISO nonlinear uncertain system is presented by the Takgi- Sugeno type fuzzy model state. We recall the principle of the sliding mode control theory then we combine the fuzzy systems with the sliding mode control technique to compute at each sampling time the control law. The control law comports two terms: equivalent control law and switching control law which has a high frequency. The uncertainty is replaced by its upper bound. Inverted pendulum and mass spring dumper are used to check performance of the proposed fuzzy robust sliding mode control scheme. Keywords: Nonlinear systems, Sliding mode, T-S fuzzy systems, Reaching law. 1 Introduction Many of the industrial plants include nonlinearities or/and uncertainties. To reach the wanted per- formances, using the classical theories, nonlinearities must be identified to calculate the appropriate controller. The robust control theories is one of the techniques that permits to reach the desired perfor- mances in presence of external or/ and internal disturbances. In addition to the stability, the tracking problem must be solved independently of uncertainties. In the literature, many methods have been de- veloped in continuous as well in discrete-time to solve the tracking problem for nonlinear systems. In the last decade many researches combine classical techniques with intelligent one, such as sliding mode with Neuronal systems or sliding mode with fuzzy systems [3][4], to benefit from the advantages of the two control techniques. The sliding mode control (SMC) was originally developed for variable structure systems in contin- uous domain. Utkin [12] gives a thorough description of the sliding mode theory in continuous time. Also, Slotine and Li [16] describe in detail continuous sliding mode controllers. At the end of the twentieth century, the research of discrete time SMC has been attracted more atten- tion, such as [6],[8],[9],[10], as for the implementation of the controller on a digital computer requires a sampling time and the assumption of an infinite switching time does not hold any more. The fuzzy systems have been combined with classical sliding mode control to provide robust stabil- ity to the fuzzy controller. The combination of the two control principles, is called fuzzy sliding mode control (FSMC), it provides an alternative to design a robust controller for nonlinear systems with uncer- tainties [15],[14]. Our contribution in this work consists in presenting a new robust fuzzy sliding mode controller based on the Takagi-Sugeno fuzzy state model for discrete nonlinear systems. This paper is organised as follow. In Section 2, we recall the discrete Takagi-Sugeno type fuzzy state model for nonlin- ear systems. Then, we describe the sufficient and necessary reaching conditions of sliding mode control for discrete nonlinear systems in the first part of the third section ten, a fuzzy sliding mode controller for discrete time of nonlinear systems is developed in the second part and tracking robust fuzzy sliding mode control law is described in the third part. The simulation results of two nonlinear systems show performances of the proposed FSMC in Section 4. Conclusions are drawn in the final section. Copyright © 2006-2008 by CCC Publications Robust Fuzzy Sliding Mode Controller for Discrete Nonlinear Systems 7 2 Problem Statement and Fuzzy Systems 2.1 Problem Statement Consider a class of discrete nonlinear SISO systems described by the following equations:    x1 (k + 1) = x2(k) xi (k + 1) = xi+1(k) xn (k + 1) = f (X (k)) + g (X (k)) u(k) y (k) = CX (k) (1) where, C=[1, 0, ..., 0], X (k) = [ x1(k) . . . xn (k) ]T ∈ Rn it is the state vector that is assumed to be observable. We note that f (X (k)) and g(X (k)) represent two discrete bounded nonlinear functions of the nonlin- ear SISO systems . They can be obtained from the continuous form by the first order discretized system using Eulers approximation. In order for (1) to be controllable, it is required that g(X (k)) 6=0. If both functions f (X (k)) and g(X (k)) in (1) are available for feedback, the feedback linearization control can be used to design a well-defined controller, which is usually given in the form: u (k) = 1 g (X (k)) ( −KT E (k)− f (X (k)) + xnd (k + 1) ) (2) where, the state vector X (k)and the desired state vector Xd (k) are defined as: X (k) = [ x1(k) . . . xn (k) ]T ∈ Rn ; Xd (k) = [ x1d (k) . . . xnd (k) ]T ∈ Rn E (k) represents the state tracking error, it is defined as: E (k) = X (k)− Xd (k), the vector K (k) =[ kn kn−1 . . k1 ]T ∈ Rn will be chosen such that all roots of the following polynomial h (s) = sn +k1sn−1 +...+ kn are situated inside the unit complex disc. In general case, the functions f (X (k)) and g(X (k)) are badly known nonlinear functions so, the control law (2) cannot be implanted. To overcome this difficulty, many approaches are used such as adaptive control, linearization around operating points, fuzzy control. . . etc. 2.2 Discrete Takagi-Sugeno type Fuzzy Systems The advantage of the T-S type fuzzy models is that their description permits the utilization of the state representation, and by consequence to exploit the maximum of the potential relative to this repre- sentation. The Takagi-Sugeno (T-S) type fuzzy model can be viewed as a natural expansion of piecewise linear partition for nonlinear systems. The nonlinear system is represented as a collection of the fuzzy IF-THEN rules, where each rule describes the local dynamics by a linear system model. The general fuzzy model is achieved by fuzzy amalgamation of the linear systems models [1][2]. The ith rule of the discrete fuzzy model has the following form: ith plant rule : IF Z1is µi1and...and Zn is µin T HEN X (k + 1) = (Adi + ∆Adi) X (k) + (Bdi + ∆Bdi) u (k) (3) where { µi j } are the fuzzy sets, Adi ∈ Rnxn and Bdi ∈ Rnxm are recpectively the ith state matrix and the input matrix, c is the number of the IF-THEN fuzzy rules; u(k) is the input vector, Z (k) =[ Z1(k) . . . Zn (k) ] are the premise variables they represent some measurable system variables, they can be chosen as a state variables. For each rule Ri is attributed a weight wi(z(k)) which depends on the grade of the membership function of the premise variables z j(k) in fuzzy sets µi j : 8 Hafedh Abid, Mohamed Chtourou, Ahmed Toumi wi (Z (k)) = n ∏ j=1 µi j (Z j (k)) ; wi (Z (k))  0 ;for i = 1, ..., c; c ∑ i=1 wi (Z (k))  0. where, µi j (Z j (k)) is the grade of the membership function of Z j (k) to the fuzzy set µi j. The discrete Takagi-Sugeno type fuzzy model is inferred as follows: X (k + 1) = c ∑ i=1 wi (Z (k)) ((Adi + ∆Adi) X (k) + (Bdi + ∆Bdi) u (k)) c ∑ i=1 wi (Z (k)) (4) The normalized weight is defined as which is presented as follow: hi(Z (k)) = wi(Z(k)) c ∑ i=1 wi(Z(k)) ; 0 ≺ hi(Z (k)) ≺ 1; i = 1, ..., c; c ∑ i=1 hi (Z (k)) = 1. The output of the discrete Takagi-Sugeno type fuzzy model for the uncertain nonlinear systems can be inferred as:    X (k + 1) = c ∑ i=1 hi (Z (k)) ((Adi + ∆Adi) X (k) + (Bdi + ∆Bdi) u (k)) y (k) = c ∑ i=1 hi (Z (k))CiX (k) (5) where Ci = [1, 0, 0, . . . , 0]. It is required that all Bdi(k) are different from zero to assure the controllability of (5). 3 Sliding mode control law and reaching conditions Sliding mode control, first appeared in the Soviet literature, it has been widely recognized as a po- tential approach to uncertain dynamical non-linear systems that are subject to external disturbances and parameter variations [12]. In sliding mode control (SMC), the control action forces the system trajectories to cross a manifold of the state space which is called the sliding surface designated by the designers [12]. The system trajectories are then constrained to the sliding surface for all subsequent time via the use of high speed switching controls. The most significant advantage of the sliding mode is robustness against changes in system parameters or disturbances. The major disadvantage associated to the sliding mode control is the chattering phenomena, because it can excite undesirable high frequency dynamics. The sliding mode control comports three modes, namely, the reaching mode (RM), sliding mode (SM), and steady-state mode (SS). Let us describe the discrete sliding mode control however, only a few researches are interested by discrete-time systems. A discrete version of SMC has a big importance when the implementation of the control is realized by numerical components which need a sampling period to compute the appropriate controller. It must be pointed out that the discrete version of SMC cannot be obtained from their continu- ous counterpart by means of simple equivalence. Among the first which are interested by SMC problem and used an equivalent form of the continuous reaching condition to give a discrete reaching condition are Dote and Hoft [5]. [S (k + 1)−S (k)] S (k) ≺ 0 (6) Milosavljevic [6] recommended the concept of the quasi-sliding mode and signalled hat condition (6) is not sufficient for a discrete sliding mode control. Robust Fuzzy Sliding Mode Controller for Discrete Nonlinear Systems 9 Sarpturk, et al. [7], used the following reaching condition. |S (k + 1)| ≺ |S (k)| (7) Furuta [8] used the equivalent form of a Lyapunov-type of continuous reaching condition to give the discrete version. V (k + 1)−V (k) ≺ 0 with V (k) = 1 2 (S (k))2 (8) Weibing Gao et al. [9] pointed out that all these forms of reaching conditions are incomplete for a satisfactory characterization of a discrete-time sliding mode. He suggests that the state trajectory of a discrete sliding mode control system must have some attributes which form the basis of the discrete sliding mode control, for more information see [9]: 3.1 Discrete fuzzy sliding mode control law For a discrete SMC the following reaching law has been chosen: S (k + 1) = S (k)−qT S (k)−ε T sgn (S (k)) , with1−qT  0, ε  0, q  0. (9) The sliding surface is defined as: S (k) = GT X (k) where GT is a constant row vector GT = [g1, ..., gn−1, 1] such that all the roots of the following polynomial are situated in the left-half open complex plane: h (s) = sn−1 + gn−1sn−2 + ... + g1 The sliding Mode control comports two terms which are:equivalent control term and switching con- trol term[3][5][7][12]. ug = ue + us (10) The equivalent control law In the first part we assume that : ∆Adi = 0nxn and ∆Bdi = [0, ..., 0]T . The switching function is defined as: S (k) = GT X (k) The ideal quasi sliding mode satisfies: S (k + 1) = S (k) = 0 We deduct : 0 = GT c ∑ i=1 hi (Z (k)) (Adi (k + 1) X (k) + Bdi ue (k)) ; k = 0, 1, ... (11) The equivalent control term is given by: ue (k) = − ( c ∑ i=1 hi (Z (k)) G T Bdi )−1 [ GT c ∑ i=1 hi (Z (k))Adi (k + 1) X (k) ] (12) We assume that condition is hold: ( c ∑ i=1 hi (Z (k)) GT Bdi ) 6= 0 The switching control law From the reaching law we can write:S (k + 1)−S (k) = −qT S (k)−ε T sgn (S (k)) S (k + 1)−S (k) = GT c ∑ i=1 hi (Z (k)) (Adi (k + 1) X (k) + Bdi u (k))−GT X (k) (13) 10 Hafedh Abid, Mohamed Chtourou, Ahmed Toumi If we compare the two latest equations we deduct the global control law: ug (k) = −z−1 [ c ∑ i=1 hi (Z (k)) ( GT Adi (k + 1) ) X (k)−(1−qT ) S(k) + ε T sgn (S (k)) ] (14) where z = ( c ∑ i=1 hi (Z (k)) GT Bdi ) From equations and we obtain the switching control term: us (k) = − ( c ∑ i=1 hi (Z (k)) G T Bdi )−1 [−(1−qT ) S (k) + ε T sgn (S (k))] (15) 3.2 Robust fuzzy sliding mode Control law Consider the discrete system in the perturbed condition. It will be described by the T-S type fuzzy model . Where ∆Adi represents system parameters variation and ∆Bdi is the external disturbance for each sub-model. We assume the matching conditions are satisfied: ∆Adi = Bdi ^ Adi and ∆Bdi = Bdi ^ Bdi where, ^ Adi is a row vector and ^ Bdi is a scalar . They should be written as: ^ Adi = [ −∆ai1 −∆ai2 . . . −∆ain ] ^ Bdi = −∆bi. Then the equation (5) becomes: X (k + 1) = c ∑ i=1 hi (Z (k)) ( AdiX (k) + Bdiu (k) + ^ AdiX (k) + ^ Bdi ) (16) However, the global control law will be expressed as: ug (k) = −z−1 [ c ∑ i=1 hi (Z (k)) ( GT Adi (k + 1) ) X (k) + Γ−(1−qT ) S (k) + ε T sgn (S (k)) ] (17) where Γ = GT Bdi ( ^ AdiX (k) + ^ Bdi ) In general case and are unknown, but their upper bound are known, so the last global control law can not be implemented. However, to over com this difficulty we replace respectively the unknown terms and by the following expressions: Ai = (√( eig ( ^ ATdiup ^ Adiup )))T ; Bi = ∆bi max. ^ Adiup = [ −∆ai1 max −∆ai2 max . . . −∆ain max ] ^ Bdiup = −∆bi max We define a new set of perturbations and control parameters as follow:. Sig = GT BdiAiX (k); Fig = GT BdiBi; The choice of Sig and Fig is done to ensure that the sign of the incremental S(k) is opposite to the sign of S(k). The global control law will be expressed as: ug (k) = −z−1 [ c ∑ i=1 hi (Z (k)) ( GT Adi ) X (k) + Qi −(1−qT ) S (k) + ε T sgn (S (k)) ] (18) where Qi = (Sig + Fig)−(Sig + Fig) sgn (S(k)) Robust Fuzzy Sliding Mode Controller for Discrete Nonlinear Systems 11 3.3 Tracking robust fuzzy sliding mode Control law The tracking problem will be transformed into the stability problem. Indeed,S (k) = 0 represents an equation whose unique stationary solution is . The tracking problem of the desired vectorXd comes back to locate inside the quasi-sliding band width the sliding surface for all sampling time. The sliding surface will be expressed as: S (k) = GT X̃ (k) where, X̃ (k) = X (k)−Xd (k) The control law will be expressed as: ug (k) = −z−1 [ c ∑ i=1 hi (Z (k)) ( GT Adi ) X̃ (k) + Qi −(1−qT ) S (k) + ε T sgn (S (k)) ] (19) 4 Illustration To illustrate the performance of the presented approach, we choose inverted pendulum and Mass spring damper which are widely used in the control literature of nonlinear system. 4.1 Inverted pendulum The equations of system in continuous form are given by (20) [18]: where, x1 is the angle in radian of the pendulum from the vertical axis; x2 is the angular velocity in rad/s; g is the gravity acceleration; m and 2l are respectively the mass and the length of the pendulum; M is the mass of the cart and u is the force applied to the cart. The nominal values of the parameters are: g= 9.81m/s2, m = 2 kg, M = 8 kg, 2l =1m.    . x1(t) = x2(t) . x2(t) = f (x1, x2) + g(x1, x2)u + d(t) f (x1, x2) = mlx22 sin x1 cos x1−(m+M)g sin x1 ml cos2 x1− 4l3 (m+M) ; g(x1, x2) = cos x1 ml cos2 x1− 4l3 (m+M) (20) The membership functions for xi ∈ ]−π/2, π/2[are: µ1i(xi) = 1− ∣∣∣ xi(k)π/2 ∣∣∣ and µ2i(xi) = ∣∣∣ xi(k)π/2 ∣∣∣ , The state matrices and input vectors for sub-systems are: Ad1 = [ 1 0.01 0.1729 1 ] ; Ad2 = [ 1 0.01 0.0936 1 ] , Bd1 = [ 0 0.0018 ] , Bd2 = [ 0 0.000052 ] ∆Ad1u p = [ 0 0 0.05229 0 ] , ; ∆Ad2up = [ 0 0 0.028 0 ] , ∆Bd1up = 0.0005; ∆Bd2up = 0.00002; We have been chosen: q = 70; T = 0.01; ε = 0.1; GT = [10 1]; The figure 1 presents the simulation results of the behavior of variable state x1(k)and S(k) of inverted pendulum for nominal system. The initial conditions are given by:X (0) = [π/3; 0]. We present by the figures2, 3, 4, 5, 6 and 7 the simulation results of the behavior of the state vari- ables x1(k), x2(k), the position and velocity error e1(k) and e2(k), the sliding surface and the control law respectively of the inverted pendulum with parameters vary of an uncertain way in time.The initial conditions are given by: X (0) = [−π/60; 0] ; GT =[15 1], q=80, ε =1.5; The function sign is replaced by the well known sat function which is defined as: { if S ≺ 1Φ ; sat = S if S º 1Φ ; sat = sgn(S) 12 Hafedh Abid, Mohamed Chtourou, Ahmed Toumi 0 50 100 150 200 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Evolution of X1(k) and S(k) (k) X1(k) S(k) Figure 1: Stabilization of X1(k) and the sliding surface S(k) 4.2 Mass spring damper The Mass-spring-damper system is described in continuous time by the following equation [17]: M1ẍ1(t) + c1x1 (t) + c2 · x1 (t) + c3x1 (t) + c4x21 (t) = ( 1 + c5 ( · x1 )3 (t) ) u (t) TheT-S fuzzy model is discribe by the following rules: Rule 1: IFx1 is M11 and x2 is M12 THEN X (k + 1) = Ad1X (k) + Bdi1u (k) Rule 2: IF x1 is M21 andx2 is M22 THEN X (k + 1) = Ad2X (k) + Bdi2u (k) Rule 3: IF x1 is M31 and x2 is M32 THEN X (k + 1) = Ad3X (k) + Bdi3u (k) Rule 4: IF x1 is M41 and x2 is M42 THEN X (k + 1) = Ad4X (k) + Bdi4u (k) For nominal values of M1, c2, c3 and c4, matrices Ai and Bi are given by: Ad1 = Ad2 = [ 1 0.01 −0.0001 0.99 ] ; Ad3 = Ad4 = [ 1 0.01 −0.0023 0.99 ] ; Bd1 = Bd3 = [ 0 0.0143 ] , Bd2 = Bd4 = [ 0 0.0056 ] ; ∆Ad1up = ∆Ad2up = [ 0 0 0 0.003 ] , ∆Ad3up = ∆Ad4up = [ 0 0 0.0007 0.003 ] , ∆Bd1up = ∆Bd3up = 0.043; ∆Bd2up = ∆Bd4up = 0.0017; The initial condition and parameters are chosen as:X (0) = [−π/60; 0] ; GT =[15 1], T=0.01; q=70, ε =0.15. We present by the figures 8and 9 the simulation results of the behavior of the state variables x1(k), x2(k), of the mass spring damper with parameters vary of an uncertain way in time. We present by the figures 10, 11, 12 and 13 the simulation results of the behavior of the position and velocity error e1(k) and e2(k), the sliding surface and the control law respectively of the mass spring damper with parameters vary of an uncertain way in time. Robust Fuzzy Sliding Mode Controller for Discrete Nonlinear Systems 13 0 1000 2000 3000 4000 5000 6000 7000 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 Evolution of X1 and X1d k X1(k) X1d(k) Figure 2: Evolution of x1 and x1d 0 1000 2000 3000 4000 5000 6000 7000 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 Evolution of X2 and X2d k X2(k) X2d(k) Figure 3: Evolution of x2 and x2d 14 Hafedh Abid, Mohamed Chtourou, Ahmed Toumi 0 1000 2000 3000 4000 5000 6000 7000 −0.01 −0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Evolution of the position error k Figure 4: Evolution of the position error 0 1000 2000 3000 4000 5000 6000 7000 −0.2 −0.15 −0.1 −0.05 0 0.05 Evolution of the speed error k Figure 5: Evolution of the speed error Robust Fuzzy Sliding Mode Controller for Discrete Nonlinear Systems 15 0 1000 2000 3000 4000 5000 6000 7000 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 Evolution of the sliding surface k Figure 6: Evolution of the sliding surface 0 1000 2000 3000 4000 5000 6000 7000 −50 −40 −30 −20 −10 0 10 20 30 40 50 Evolution of the control law U k Figure 7: Evolution of the control law 16 Hafedh Abid, Mohamed Chtourou, Ahmed Toumi 0 200 400 600 800 1000 1200 1400 1600 1800 2000 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 Evolution of X1 and X1d k X1(k) X1d(k) Figure 8: Evolution of x1 and x1d. 0 200 400 600 800 1000 1200 1400 1600 1800 2000 −0.5 0 0.5 1 1.5 Evolution of X2 and X2d k X2(k) X2d(k) Figure 9: Evolution of x2 and de x2d. Robust Fuzzy Sliding Mode Controller for Discrete Nonlinear Systems 17 0 20 40 60 80 100 120 140 160 180 200 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 Evolution of the position error k Figure 10: Evolution of the position error 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 0.5 1 1.5 2 2.5 Evolution of the speed error k Figure 11: Evolution of the speed error 18 Hafedh Abid, Mohamed Chtourou, Ahmed Toumi 0 200 400 600 800 1000 1200 1400 1600 1800 2000 −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 Evolution of the sliding surface k Figure 12: Evolution the sliding surface 0 200 400 600 800 1000 1200 1400 1600 1800 2000 −20 0 20 40 60 80 100 Evolution of the control law U k Figure 13: Evolution of the control law Robust Fuzzy Sliding Mode Controller for Discrete Nonlinear Systems 19 5 Conclusions In this paper we present a robust fuzzy sliding mode controller for discrete nonlinear systems. First, we recall the discrete Takagi-Sugeno type fuzzy model, then the principe of the sliding mode control in discrete time. The uncertainty are assumed to be verifie the matching conditions. We develop a robust controller based on the sliding mode and the dynamic T-S fuzzy state model. The uncertainties are replaced bye the bigger eigen-value of the upper bound matrices of uncertainties. The expressions in discrete time of both equivalent control term and hitting term are developed. The tracking control law is developed. Simulation results for inverted pendulum and mass spring damper with parameters variation show the performance of the proposed control law. Bibliography [1] Michio Sugeno "On Stability of fuzzy Systems expressed by rules with singleton consequents, (IEEE Transaction on Fuzzy Systems,Vol 7 N 2 Feb 1999). [2] T.Takagi and M.Sugeno, Fuzzy identification of systems and its applications to modeling and con- trol IEEE trans syst. Man,Cybern. Vol15. 116-132 Jan/Feb1985. 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Hafedh Abid1,3, Mohamed Chtourou2, Ahmed Toumi3 1Institut supérieure des études technologiques de Sfax Laboratoire d’Automatique, Génie Informatique et Signal Cité Scientifique, BP 48, 59651 Villeneuve d’Ascq, France 2Unité de Commande Intelligente, design et optimisation des Systèmes complexes(ICOS) ENIS, B.P. W, 3038 sfax, Tunisie 3Unité de Procédés Industriels Unité de Commande Automatique (UCPI) ENIS,B.P. W, 3038 sfax, Tunisie E-mail: Hafedh.abid@isetso.rnu.tn Mohamed.chtourou@enis.rnu.tn Ahmed.toumi@sta-tn.com Received: June 13, 2007