Limit cycle oscillator in nonlinear systems with multiple time delays

Abstract

This paper concerns with the problem of generation stable limit cycles in a class of nonlinear time-delay systems with multiple time delays. With the help of the Lyapunov–Krasovskii functional approach, a state-feedback controller is constructed based on the backstepping technique. According to the recursive procedure of the backstepping technique, at first, the desired limit cycle is generated in the second-order subsystem by utilizing the Lyapunov theorem analysis of the positive limit sets. Then, this method is expanded for higher-order systems. By introducing an appropriate Lyapunov-Krasovskii functional, the complexities raised by unknown time-delays are solved in the control design process. Based on this scheme, a delay-independent controller is explicitly designed which does not need the precise knowledge of time delays. The proposed method rigorously guarantees the practical stability of the closed-loop system and also ensures the convergence of the phase trajectories of the closed-loop system to the target limit cycle. Finally, to prove the theoretical achievement and also to illustrate the effective performance of the proposed approach, the simulation results are provided for several examples.