Robust sliding mode control of discrete fractional difference chaotic system

In this study, discrete fractional difference chaotic system (DFDCS) is developed utilizing Caputo–delta fractional difference operator on discrete domain, and the discrete fractional sliding mode control (SMC) problem associated with this system is investigated. Initial, DFDCS is established on discrete domain with an exact numerical formula in order to avoid the numerical errors arising from the numerical discretization of the continuous case. Then, a discrete fractional switching surface function is constructed, and the stability of discrete fractional sliding mode dynamic is explored. Based on the discrete reaching law, a discrete fractional SMC technique is proposed such that the state trajectories of DFDCS in the nominal case can be driven to quasi-sliding mode (QSM) band without escape. In addition, for the uncertain discrete fractional difference chaotic system (UDFDCS) with modeling uncertainty and external disturbance, the discrete fractional adaptive SMC strategy is developed to enable the state trajectories of this system to reach QSM band and remain in it. Finally, numerical simulations employing the exact numerical formula of DFDCS are conducted to demonstrate the efficiency of the proposed discrete fractional SMC approach.