Model-based robust chaotification using sliding mode control

References (2)

1. the chaotified system with the model-based method in [32,33] are shown. In Figureg–i, z1versus z2, z, and z4of the chaotified system with the proposed sliding mode control method are shown. In Figure3j, chaotifying control input in (50) for the proposed method is presented. In Figure3d–3f, it is observed that the chaotified system with the model-based methods in [32,33] exhibits different behavior than the reference chaotic system due to the effect of the uniformly distributed noise. As seen in Figure3g–i, the chaotified system with the proposed method reaches the chaotic manifold despite the noise and slides on it thereafter. Furthermore, in order to show the effectiveness of the proposed method, the system in (48) is subjected to uniformly distributed random noise δ(t) in the interval [d, d], where d and d are chosen randomly between [-2,2] for 100 trials. For the model-based method [32,33] just 7 of the 100 trials have Lyapunov exponents λ1> 0 in the interval of [0.0145,0.1543], λ2∼ = 0 in the interval of [-0.0049,0.0061], λ 3 of [-0.6464,-0.5121], and λ4 4-dimensional systems [46], whereas the proposed method copes with noises and after a finite transient time it exhibits chaotic behavior for all trials. Results and conclusion A sliding mode control-based robust chaotification scheme has been introduced for model-based chaotification. The scheme can be applied to any continuous time single input controllable linear and input state linearizable nonlinear systems subject to parameter uncertainties, nonlinearities, noises, and disturbances that are all additive to the input and can be modeled as an unknown function but bounded by a known function. It is assumed that a reference chaotic system exists in the normal form and the designed dynamical state feedback control law forces the system to match the reference chaotic system in finite time irrespective of the mentioned uncertainties, noises, and disturbances, as provided by the developed sliding mode control scheme. The matching of the considered system to the reference chaotic system is always achieved in finite time, which can be made arbitrarily small by modifying a parameter changing the control input. Several simulations have demonstrated the robustness and effectiveness of the chaotification scheme. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, New York, Springer-Verlag, 2003.
2. R.L. Devaney, An Introduction to Chaotic Dynamical Systems, Colorado, Westview Press, 2003. S