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SLIDING MODE CONTROL USING RADIAL BASIS FUNCTION NEURAL NETWORKS Tran Quang Thuan, Lecturer, PTIT-HCM, Duong Hoai Nghia, Lecturer, HCM City University of Technology, and Dong Si Thien Chau, Lecturer, Ton Duc Thang University Abstract This paper considers the problem sliding mode control of nonlinear dynamical system using radial basis function neural networks This paper presents sliding mode control (SMC) approach, nonlinear system identification using radial basis function (RBF) neural networks approach The application of the nonlinear system identification results to design sliding mode control law for a nonlinear dynamic plant will be discussed in this paper Simulation results are presented I INTRODUCTION Earliest notion of sliding mode control strategy was constructed on a second order system in the late 1960s by Emelyanov [5] Since then, the theory has greatly been improved by researchers To design a sliding mode control system, we must have exactly model of the plant To the concrete, designers not always have exactly model of the plant To decide this problem, we propose to identify model of the control plant base on RBF neural network In comparison with traditional nonlinear identification technique, the designers which use this method don’t need to determine the structure of the model The application of the model to design sliding mode control law for a nonlinear dynamic plant will be discussed in this paper The remainder of the paper is organized as follows The second section presents nonlinear system identification using radial basis function neural networks SMC using RBF neural network are presented in the third section The following section describes the dynamic model of the plant and the result of the simulations References constitute the last part of the paper II NONLINEAR SYSTEM IDENTIFICATION USING RADIAL BASIS FUNCTION NEURAL NETWORK problem identifying the system (II.1) from inputoutput data using a RBF model: n − X −Z yˆ k +1 = ∑ θi e k i /σi (II.3) i =1 where, yˆ k is the output of the model, X k is defined as (II.2), n is the number of RBF functions, θ i are linear weights, Zi and σi respectivery, the centers (reference points) and the scaling factor of the RBF [3], [4] For identification purpose, we rewrite (II.3) as: yˆ k +1 = ΦTkθˆk (II.4) T where, θˆk = [θ1 ,θ , ,θ n ] is the linear weights vector at time k, and Φ k = e  − X k − Z1 / σ1 , , e − X k − Zn /σ n   T (II.5) is the regressive vector To identify the linear weights, we can use the stochastic approximation approach [8]: θˆk +1 = θˆk + Fk +1Φ k [ yk +1 − ΦTkθˆk ] with Fk +1 = αk δ k + ΦTk Φ k (II.6) (II.7) Here, α k is a stochastic approximation parameter which satisfies: ∞ ∑α k =∞, ∑α p k < ∞ , p≥2 k =1 ∞ lim sup(α k−1 − α k−−11 ) < ∞ , and k →∞ (II.8) k =1 δ k is the Tikhonov parameter which satisfies the following conditions: δ k → + , δ k δ k +1 > , δ k > (II.9) k →∞ III SMC USING RBF NEURAL NETWORK Consider a nonlinear system: y k +1 = f ( X k ) (II.1) X k = [ yk , , yk − n y +1 , uk , , uk − nu +1 ]T (II.2) where, yk is the output, uk is the input, nu and n y are integers In this paper, we consider the Let a nonlinear system be defined as  x ( n ) = f ( X ) + g ( X ).u  y = x (III.1) Here, X = [ x x& x ( n −1) ]T is state vector, y is the output, u is control signal, n is number of the state variables The functions f = f ( X ) , g = g ( X ) are not exactly known, but the extent of the imprecision on f, g are bounded by known: f ≤ f ≤ f max , < g ≤ g ≤ g max (III.2) Consider two of sliding mode control problems: - Tracking control: the control problem is to get feedback control law u = u(r, X) so that y→ r with r is a reference signal - Regulation control: the control problem is to get feedback control law u = u(X) so that y → when t→ ∞ A time varying surface S(t) is defined in the state space R(n) by equating the variable S(X;t), defined below, to zero S = x ( n −1) + an − x ( n − 2) + + a1 x (1) + a0 x (III.3) with a0, a1,…, an-2 are constans which the specificity function of (III.3): p n −1 + an − p n − + + a1 p + a0 = (III.4) has all roots which real of them are positive In this paper, regulation control is presented Consider a nonlinear system:  x&1 = x2   x&2 = f ( x) + g ( x).u u= −1   x2 + f ( X ) + k sign( S )  g ( X ) τ  (III.13) so that S& = −k sign( S ) (III.14) where, k is positive constant IV SIMULATION A Introducion about plant to control In this study, a couple double pendulum systems, which are illustrated in Fig.1, is used to elaborate performance of the method discussed The differential equations characterizing the behavior of the system are given in (IV.1), in which the angular positions x1 and x2, the angular velocities x3 and x4 define the state vector The control inputs, which are denoted by u1 and u2, are provided to the relevant pendulum by the base servomotors The model introduced in this section has been studied by Spooner and Passino [1], Efe, Kaynak, Yu and Wilamowski [2], Efe [7] The parameters of the plant are given in Table 1, which states that as b sup( f1 − fˆ1 ) k > sup( f − fˆ ) 2 (IV.3) (IV.4) (IV.5) (IV.6) Fig.8a: Signals ẏ1 and ẏ2 of ideal SMC system The simulation results are given in Fig.7, Fig.8, Fig.9 and Fig.10 The parameters of control system are time-constants τ1 = τ2 = 0.2 sec, positive constants k1 = k2 = 15 and initial conditions [x1 x2 x3 x4]T = [-π/4 π/7 0]T Under these conditions, phase orbit motions depicted in Fig 10 are obtained The simulation results of SMC system using RBFNN and ideal SMC system, which use exactly f1(X) and f2(X), are analogous Fig.8b: Signals ẏ1 and ẏ2 of ideal SMC system using RBFNN V REFERENCES Preriodicals: Fig.9a: Sliding surfaces of ideal SMC system [1] Jeffrey T.Spooner and M Passino, “Decentralized Adaptive Control of Nonlinear Systems Using Radial Basis Neural Networks”, IEEE Transactions on Automatic Control, vol.44, No.11, pp.2050-2057, 1999 [2] M.Önder Efe, Okayay Kaynak, Xinghuo Yu, Bogdan M.Wilamowski, “Sliding Mode Control of Nonlinear Systems Using Gaussian Radial Basis Function Neural Networks”, IEEE Transactions on Neural Networks, pp.474479, 2001 [3] Hui Peng, Tohru Ozaki, “A Parameter Optimization Methode for Radial Basis Function Type Models”, IEEE Transactions on Neural Networks, Vol.14, No.2 (2003), pp.432-438 [4] Chu Kiong Loo, Mandava Rajeswari, M.V.C Rao, “Novel Direct and Self – Regulating Approaches to Determine Optimum Growing Multi-Expert Network Structure”, IEEE Transactions on Neural Networks, Vol.15, No.6 (2004), pp.1378-1395 Books: [5] Emelyanov, S.V “Variable structure control sytems”, Moscow, Nauka, 1967 [6] H Khalil, “Nonlinear Systems”, 2002 Papers fom Conference Proceedings Fig.9b: Sliding surfaces of SMC system using RBFNN [7] M.Önder Efe, “Variable Structure Systems Theory in Training of Radial Basis Function Neurocontrollers – Part II: Applications”, NIMIA-SC2001 – 2001 NATO Advanced S1=0 S1>0 S10 S2