Designing Fractional Order Sliding Mode Control Based on Nonlinear Adaptive Law for Stabilization and Synchronization of a Class of Fractional Order Chaotic Systems

In this paper, the problem of stabilization and synchronization of Lorenz and Chua chaotic in the presence of uncertainty using fractional order sliding mode control strategy based on nonlinear adaptation law has been investigated. Lorenz and Chua systems denote third order dynamics models which are chaotic for certain parameters. The proposed control law is composed of two prats sliding mode control and adaptive control law. Firstly, by supposing that instantaneous information of nonlinear part of chaotic system is not available, a linear regressor equation including an unknown section has been used. Using Lyapunov stability theorem and based on fractional calculus, adaptation law is developed to instantaneous estimation of unknown part. Moreover, by defining based on error signals and realizing exponential reaching law for insuring closed-loop stability, the sliding mode control law including equivalent and switching control has been derived. Eventually, the final control law has been derived by synthesizing sliding mode control and adaptive laws. The important aspect of the proposed approach is ability to encounter unstructured uncertainties and nonlinear effects of chaotic systems dynamic and guiding the state variables into sliding surface for arbitrary initial conditions. The performance of the proposed algorithm has been evaluated by realizing the stabilization problem of chaotic Lorenz system and synchronization of chaotic Lorenz and Chua systems.