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European Journal of Scientific Research ISSN 1450-216X Vol.14 No.3 (2006), pp. 326-332 © EuroJournals Publishing, Inc. 2006 http://www.eurojournals.com/ejsr.htm Model-Based Adaptive Chaos Control using Lyapunov Exponents Amin Yazdanpanah Goharrizi Department of Electrical Engineering K. N. Toosi University of Technology Tehran, Iran Mehdi Semin Department of Electrical Engineering Tabriz University, Tabriz, Iran E-mail: yazdanpanah@ee.kntu.ac.ir Abstract A model-based approach to adaptive control of chaos in non-linear chaotic discrete time systems is presented. In the case of unknown or time varying chaotic plants, the Lyapunov exponents may vary during the plant operation. In this paper, an effective adaptive strategy is proposed for on-line identification of Lyapunov exponents. The control aim is that the plant output changes in accordance with the output of the linear desired model. Also, a nonlinear observer for estimation of the states is proposed. Simulation results are provided to show the effectiveness of the proposed methodology. 1. Introduction The analysis and control of chaotic behavior in dynamical systems has been widely investigated in recent years [1], [2], [3], [4] and [5].Also, Lyapunov exponents have been used to characterize and quantify the chaoticity of complex dynamical systems [6], and the computation of the Lyapunov exponents for nonlinear dynamical systems is an effective tool in this respect [7]. In [8], a model-based approach for anticontrol of some discrete-time systems is proposed and in this paper, a reverse method for control of chaotic systems is presented and also, a new method to adaptive control of chaos via adaptive calculation of Lyapunov exponents is introduced. The adaptive calculation of Lyapunov exponents proposed in [9-10-11], greatly facilities the design of adaptive chaos control. Thus, a generalized adaptive algorithm recursive least square for estimation of Lyapunov exponents is developed when the parameters of the system change abruptly. We use an efficient QR based method for the computation of Lyapunov exponents [12]. Then, if the maximum Lyapunov exponent becomes positive it's indicates the chaotic behavior and the control aim is that the plant output changes in accordance with the output of the linear desired model. So, the behavior of the closed-loop system depends on the linear model and it can be periodic or tends to zero after controlling. With the above strategy and adaptive calculation of Lyapunov exponents an efficient methodology for adaptive chaos control is presented. Also, a nonlinear observer is proposed when the sate of nonlinear chaotic plant are not available. Finally, simulation results for Henon map with time varying parameters are provided to show the effectiveness of the proposed methodology.
Model-Based Adaptive Chaos Control using Lyapunov Exponents 327 2. Adaptive Estimation of Lyapunov Exponents Consider a nonlinear discrete-time dynamical system given by n ⎪⎧ x( k + 1) = f [ x( k)] x( k) ∈ R ⎨ ⎪⎩ x0 − given (1) Where f is assumed to be continuously differentiable, at least locally in a region of interest and k = 0, 1,... is the number of the sampling instants. Each of the scalar state variables in the nonlinear discrete-time dynamical system given by equation (1), can be rewritten in the following form T x j ( k + 1) = ϕ 1 j ( k) θ1 + ϕ2 j ( k) θ 2 + ... + ϕij ( k) θi = ϕ j ( k) θ j k = 0, 1,... j = 1,..., n , , Where, x j are the scalar state variables, θ ij s are parameters to be determined, and ϕ ij s are known functions that may depend on other known variables. The vectors T ϕ ( k) = [ ϕ ( k) ϕ ( k) ... ϕ ( k)] j 1 j 2 j ij θ j = ( θ1 j θ2 j ... T θij ) (2) have also been introduced. By invoking the recursive least square (RLS) algorithm, the parameters of the model in (2) can be updated recursively [13]: ˆ ˆ T θ ( ) ( 1) ( )[ ( ) ( ) ˆ j k = θ j k − + K k x j k −ϕ j k θ j ( k −1)] (3) Where T −1 K ( k ) = P( k −1) ϕ j ( k )[ λI + ϕ j ( k ) P( k −1) ϕ j ( k )] (4) And T P ( k ) = [ I − K ( k ) ϕ j ( k )] P ( k − 1) / λ (5) And, λ is the forgetting factor and I is an i × i identity matrix. Let J k be the Jacobian matrix of (1) which is updated at each iteration by adaptive calculation of the system parameters. It is shown that the QR-factorization of the matrix product J m J m− 1...J1 gives the Lyapunov exponents of the plant in equation (1). This process is started with Q 0 = I , and is as follows: qr[ J J ... J ] = qr[ J J ... J ( J Q )] = qr[ J qr[ J qr[ J m m m m J J J m−1 m−1 m−1 m−1 ... J ...( J Q )][ R R ] = ... ... J 1 3 i ( J Q )][ R ] = 3 Q 2 2 i−1 1 ][ R m 2 i−1 R m−1 1 1 i− 2 ... R 2 2 R 1 1 0 ] = ... Qm [ Rm ... R2 R1] = Qm R (6) Where, qr[.] denotes the QR-factorization process. Starting with J 1 , at each step, a pre- B J Q followed by a QR-factorization of = J Q − = Q R i = 1, 2,..., m is multiplication i = i i−1 performed. Matrix R is the product of the matrices ... R2R1 Bi i i 1 i i, Rm obtained in this sequential manner. Furthermore, each of the diagonal elements of R is simply the product of the corresponding diagonal elements of all the , s . Hence, approximation to the n Lyapunov exponents are provided as: R i m 1 λ j = lim ∑ ln Ri ( j, j) j = 1, 2,..., n m→∞ m i= 1 (7) Since the Jacobian matrix of (1) is updated at each iteration, the Lyapunov exponents of the system are calculated adaptively at each iteration.
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