Model-based robust chaotification using sliding mode control

Chaos is a complex behavior of dynamical nonlinear systems that is undesirable in most applications and should be controlled; however, it is desirable in some situations and should be generated. In this paper, a robust chaotification scheme based on sliding mode control is proposed for model based chaotification. A continuous time single input observable system is considered such that it is subject to parameter uncertainties, nonlinearities, noises, and disturbances, which are all additive to the input and can be modeled as an unknown function but bounded by a known function. The designed dynamical state feedback control law forces the system to match a reference chaotic system in finite time irrespective of the mentioned uncertainties, noises, and disturbances, as provided by the developed sliding mode control scheme. Simulation results are provided to illustrate the robustness of the proposed scheme against parameter uncertainties and noises. The results are compared with those of other model-based methods and Lyapunov exponents are calculated to show whether the closed-loop control systems exhibit chaotic behavior or not.

Anahtar Kelimeler:

Anticontrol, dynamical feedback, robust chaotification, sliding mode control

Model-based robust chaotification using sliding mode control

Keywords:

Anticontrol, dynamical feedback, robust chaotification, sliding mode control,

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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< 0 in the interval
of [-0.6464,-0.5121], and λ4< 0 in the interval of [-1.014,-1.0018], which is the sign of chaotic behavior for
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.
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