Adaptive control and synchronization of a class of chaotic systems in which all parameters are unknown

This paper proposes an adaptive algorithm for the control and synchronization of a class of second order chaotic systems which exact dynamics or parameters are unknown in priori. The proposed control scheme includes a feedback controller and a feedforward compensator. Both the gains of the controller and the compensator are updated by an adaptation algorithm derived from Model Reference Adaptive Control (MRAC) theory. In the proposed approach, the optimal adaptation gains are identified using the NeIder-Mead simplex algorithm. This algorithm does not require the derivatives of the performance index to be optimized, and is therefore particularly applicable to complex systems or problems with undifferentiable elements, discontinuities or uncertainties. The feasibility and effectiveness of the proposed approach are demonstrated by way of numerical simulations using general Duffing´s systems for illustration purposes.