Synchronisation of 6D hyper-chaotic system with unknown parameters in the presence of disturbance and parametric uncertainty with unknown bounds

In this paper, adaptive-sliding mode control method is proposed for synchronisation of two 6D hyper-chaotic systems in the presence of external disturbance and parametric uncertainty and unknown parameters in the slave system. In the first section of this paper, two 6D integer order hyper-chaotic systems in the presence of external disturbance signal, parametric uncertainty and unknown parameters in the slave system are studied. In the second part of this paper, after identifying chaos in fractional order dynamic of the mentioned system, synchronisation of two 6D fractional derivative hyper-chaotic systems in the presence of external disturbance signal, parametric uncertainty and unknown parameters in the slave system is investigated, in which fractional order Riemann-Liouville derivative is used; a new fractional order sliding surface is defined for the hyper-chaotic system to determine the proper active control. Proper adaptive control laws are used to estimate the uncertainty bound, unknown disturbance signal and system parameters. Stability of the closed-loop control system is proved using Lyapunov theory in both modes. Simulation results in MATLAB show the desired application of the proposed controllers in the presence of disturbance and parametric uncertainty.