Sliding mode-based synchronisation and control of chaotic systems with parametric uncertainties

In this paper, we propose a result on controlling chaos using the sliding mode control method. We show that for the class of chaotic systems that can be stabilised using a smooth feedback controller, a sliding surface can be easily constructed based on the Lyapunov theory. Moreover, it is shown that if the states are confined to the sliding surface, then the originally chaotic trajectories will slide along a stable manifold towards the equilibrium. In addition to stabilisation, we show that the proposed controller can also lead to synchronisation of two chaotic systems when the problem is regarded as trajectory tracking. We also prove that the proposed controller becomes robust to parametric uncertainties by increasing the gain. Besides, the unwanted chattering phenomenon can be reduced by adaptively tuning the sliding gain. All these results are confirmed through numerical simulations on a set of jerk systems.