Limit Cycle Synchronization of Nonlinear Systems with Matched and Unmatched Uncertainties Based on Finite-Time Disturbance Observer

This paper focuses on the limit cycle control of uncertain nonlinear systems in the presence of both matched and unmatched uncertainties. For this purpose, first, a virtual nonlinear system is constructed which has the desired limit cycle in its phase trajectories. The Lyapunov stability theorem for positive limit sets is utilized to confirm the stability of the created limit cycle in the virtual system. Next, by using a modified nonsingular terminal sliding mode control method, a robust controller is designed to synchronize the trajectories of the actual nonlinear system with the corresponding virtual one in a finite time. It is assumed that the actual system suffers from both matched and unmatched uncertainties and/or unknown external disturbances. A nonlinear terminal sliding surface is designed with the aim of a finite-time disturbance observer to tackle these unknown terms. The finite-time Lyapunov stability theorem is utilized to confirm the stability and robustness of the designed control law. Through simulation results, the finite-time stabilization of the synchronization errors and appropriate performance of the proposed control law are verified.