Optimal synchronization of non-smooth fractional order chaotic systems with uncertainty based on extension of a numerical approach in fractional optimal control problems

In this paper, a control mechanism is presented for optimal synchronization of two non-smooth fractional order chaotic systems with parametric uncertainty based on nonlinear fractional order proportional derivative (NLFPD) controller combined with optimal periodic control signals. Unlike synchronization methods based on FPID controllers, in this approach, optimal tuning of the NLFPD controller and determination of optimal periodic control signals for synchronization process are presented in the form of non-smooth fractional order optimal control (FOC) problems. Optimal periodic control signals are demonstrated as a generalized expansion in the sense of Fourier expansion that is used to accelerate the synchronization process. Using the generalization of a numerical method in nonlinear optimal control problems and the Grunwald-Letnikov(GL) fractional derivative definition, the synchronization problem based on the suggested mechanism is transformed into the form of a smooth FOC problem. By defining a base-time soft switch, a supervisory approach is added to the proposed control mechanism for desired and robust performance against parametric uncertainty in the slave system. Finally, to illustrate the proposed control mechanism, the synchronization of two identical fractional order Chua systems with simulation results is presented. The Results show that using the suggested control mechanism, the synchronization is fast and robust against parametric uncertainty in Chua slave system.