Adaptive arbitrary time synchronisation control for fractional order chaotic systems with external disturbances

In this paper, adaptive synchronisation of fractional order chaotic systems with external disturbances is studied. For this purpose, first, a new integer order system is introduced and its convergence to zero exponentially within an arbitrary time is proven. Using this system, a novel adaptive exponential arbitrary time sliding mode controller is established to synchronise the fractional order chaotic systems. Then, to estimate the disturbance terms of the fractional systems an exponential arbitrary time disturbance observer is developed. The use of the disturbance observer can lead to simple controller design, low chattering control input, and more accurate response. Finally, a disturbance observer based sliding mode control is presented, that by incorporating the disturbance estimation in controller design ensures synchronisation of chaotic systems. The exponential arbitrary time synchronisation and the stability of the closed loop control system are confirmed using the Lyapunov theory. Simulation results on synchronisation of different fractional order systems, validate the proposed control schemes. It is noteworthy that the introduced adaptive sliding mode controllers can be used for controlling a wide variety of fractional order systems.