Adaptive Sliding Mode Control for Synchronization of a Fractional-Order Chaotic System

Abstract

This paper proposes a three-dimensional autonomous chaotic system which displays some interest dynamical behaviors such as invariable Lyapunov exponent spectrums and controllable signal amplitude. The corresponding fractional version of the proposed system is obtained. A single state controller for synchronization of this fractional-order chaotic system is developed based on the techniques of sliding mode control and adaptive control. Numerical simulations are provided to demonstrate the feasibility of the presented synchronization method.

Theoretical and practical applications of fuzzy fractional integral sliding mode control for fractional-order dynamical system

“…Case (ii): Two different fractional orders (α = β = 1). If α = 0.91 and β = 0.97, then the projection of the attractors between the synchronized systems (28) and (29) are depicted in Fig. 6.…”

“…4. Further, the time response between the states of (28) and (29) and their synchronization errors are depicted in Fig. 5.…”

“…If α = β = 0.97, then the projection of the synchronized attractors of the drive system (28) and response system (29) are depicted in Fig. 4.…”

“…Assume that the fractional orders α, β ≥ 0.87, the parameters μ = 2, γ = 1 and the initial conditions of the drive system (28) and the response system (29) are (x 1 (0), x 2 (0), x 3 (0)) = (1, 2, −3) and (y 1 (0), y 2 (0), y 3 (0)) = (−1, −2, 0), respectively.…”

“…The main advantages of the SMC are the fast response, low sensitivity to external disturbances, robustness to the plant uncertainties, and easy realization. Due to these advantages, the SMC strategy has been successfully applied to control and synchronize the fractional order dynamical systems, see [10,[20][21][22][23][24][25][26][27][28][29][30]. However, one major drawback of the conventional SMC approach is the high frequency of control action called chattering.…”