Stabilizing unstable periodic orbit of unknown fractional-order systems via adaptive delayed feedback control

TL;DR

This paper introduces an adaptive delayed feedback control method for stabilizing unstable periodic orbits in unknown fractional-order chaotic systems, ensuring robustness and faster convergence compared to previous methods.

Contribution

It proposes a novel adaptive control framework using Lyapunov and sliding mode techniques that requires only the orbit period and handles unknown parameters.

Findings

Effective stabilization of fractional-order chaotic systems demonstrated.

Outperforms previous linear feedback methods in reducing steady-state error.

Robust against system uncertainties and external disturbances.

Abstract

This article presents an adaptive nonlinear delayed feedback control scheme for stabilizing the unstable periodic orbit of unknown fractional-order chaotic systems. The proposed control framework uses the Lyapunov approach and sliding mode control technique to guarantee that the closed-loop system is asymptotically stable on a periodic trajectory sufficiently close to the unstable periodic orbit of the system. The proposed method has two significant advantages. First, it employs a direct adaptive control method, making it easy to implement this method on systems with unknown parameters. Second, the framework requires only the period of the unstable periodic orbit. The robustness of the closed-loop system against system uncertainties and external disturbances with unknown bounds is guaranteed. Simulations on fractional-order duffing and gyro systems are used to illustrate the effectiveness of the theoretical results. The simulation results demonstrate that our approach outperforms the previously developed linear feedback control method for stabilizing unstable periodic orbits in fractional-order chaotic systems, particularly in reducing steady-state error and achieving faster convergence of tracking error.