Finite time chattering-free control of chaos via smooth second order sliding mode

The chaos phenomenon has been observed in many natural and manmade systems. It can potentially direct a system to unstable condition, undesirable performance, and terrible situations. In many practical applications, in order to improve performance of a system and avoid undesirable situations caused by chaos, the system must be controlled in way which makes it able to remove the chaos. Moreover, the control signal must be smooth, since high frequency switching caused by chattering effect can be destructive for some control devices. This paper presents a finite time sliding mode control strategy to control chaotic systems. The main objective is controlling chaos smoothly without chattering phenomenon. The sliding manifold is constructed using Lyapunov function and is attained in finite time. It is proved that if the states are confined to the sliding surface, then the chaotic trajectory will slide toward the origin. A controller is designed, based on smooth second order sliding mode control, so that there is no chattering in the states of the system, and the control input is a smooth signal, also finite time convergence is attained. An illustrative example is given to demonstrate the effectiveness of the proposed sliding mode controller.