Robust Control and Synchronization of Chaotic Fractional-order Permanent Magnet Synchronous Machine

Permanent magnet synchronous motors (PMSM) show chaotic phenomena under certain working conditions. In this paper, both fractional order (FO) robust control via single-variable feedback and FO robust synchronization via linear feedback are achieved by the use of the FO-PMSM chaotic model with uncertainties. The proposed method decomposes the nonlinear fractional-order system (FOS) into a linear component and a nonlinear component. The linear component represents the nominal system, and the nonlinear component stands for the perturbation of the nominal system. The robustness of the system is investigated based on the study of the parameters or structural uncertainties of the nonlinear component. The FO Lyapunov matrix differential equation stability theory is proposed to obtain robust stability results of linear FOS with the order of 0<alpha<1. Numerical simulations are performed to illustrate the effectiveness of the proposed design. The results show that the performance of the controlled system is stable at the equilibrium point within a very short transition time and the synchronization error tends to zero very quickly. In addition, the stability theory based on FO Lyapunov matrix differential equation can be extended to test the stability of other fractional order systems.