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330 Amin Yazdanpanah Goharrizi and Mehdi Semin By employing the on-line Lyapunov exponents calculation strategy presented in section 2, figure 3 shows the simulation results for the given parameters variations. Figure 3: Henon Lyapunov exponents. Now, at each iteration the Lyapunov exponents are calculated with the RLS algorithm. Hence, if the maximum Lyapunov exponent becomes positive controller is applied to the system as discussed in pervious section. First the eigenvalues of matrix A is chosen as 1 , 2 1 = λ on the unit disk, so the output (state x) becomes periodic. The result is shown in figure 4. Figure 4: The closed loop response. The other simulation is performed as a regulation problem which the eigenvalues of matrix A is chosen as 1 , 2 0. 5 = λ . The output of the Henon map (state x) is shown in figure 5 and figure 6 shows the estimation error for nonlinear observer.
Model-Based Adaptive Chaos Control using Lyapunov Exponents 331 Figure 5: The closed loop response as a regulation problem. Figure 6: estimation error. 6. Conclusions In this paper, an adaptive control of chaos via Lyapunov exponents is presented. A model-based approach to adaptive control of chaos in chaotic discrete time systems is presented. A generalized RLS algorithm for adaptive estimation of Lyapunov exponents in the case of abrupt parameters change is employed. Also, when the sates of the plant are not accessible a nonlinear observer design methodology is proposed. Finally, simulation results are provided to show the effectiveness of the proposed methodology. Acknowledgements This work was supported by the complex systems research center in K. N. Toosi University of Technology.
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