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In this paper, a robust adaptive sliding mode controller is proposed. Under the existence of external disturbances, modified hybrid projective synchronization (MHPS) between two identical and two nonidentical fractional-order complex chaotic systems is achieved. It is shown that the response system could be synchronized with the drive system up to a nondiagonal scaling matrix. An adaptive controller and parameter update laws are investigated based on the Lyapunov stability theorem. The closed-loop stability conditions are derived based on the fractional-order Lyapunov function and Mittag-Leffler function. Finally, numerical simulations are given to verify the theoretical analysis.

Under the existence of model uncertainties and external disturbance, finite‐time projective synchronization between two identical complex and two identical real fractional‐order (FO) chaotic systems are achieved by employing FO sliding mode control approach. In this paper, to ensure the occurrence of synchronization and asymptotic stability of the proposed methods, a sliding surface is designed and the Lyapunov direct method is used. By using integer and FO derivatives of a Lyapunov function, three different FO real and complex control laws are derived. A hybrid controller based on a switching law is designed. Its behavior is more efficient that if the individual controllers were designed based on the minimization of an appropriate cost function. Numerical simulations are implemented for verifying the effectiveness of the methods.