Limit cycle oscillator in nonlinear systems with multiple time delays. (December 2021) - Hakimi, A.R.; Azhdari, M.; Binazadeh, T.
Highlights: This paper studies the design of limit-cycle oscillators for time-delay systems based on the Lyapunov approach, for the first time. The considered time-delay systems are nonlinear with multiple time delays. The limit cycle oscillator is shaped using an extended Lyapunov theorem for stability analysis of positive limit sets. The Lyapunov-Krasovskii theorem is employed to handle the difficulty raised by 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.