[Paper Review] Synchronization of Uncertain Fractional-Order Duffing-Holmes Chaotic System via Sliding Mode Control

S. Hassan HosseinNia, Reza Ghaderi+2 more|arXiv (Cornell University)|Jun 12, 2012

Chaos control and synchronization17 references3 citations

TL;DR

This paper proposes a sliding mode control (SMC) strategy with a modified sliding surface and saturation function to synchronize uncertain fractional-order Duffing-Holmes chaotic systems. The method ensures robust synchronization despite system uncertainties, and numerical simulations confirm its effectiveness in achieving fast, stable synchronization of the chaotic systems under study.

ABSTRACT

In this paper, a sliding mode controller is designed to synchronize a chaotic fractional-order system. To construct a corrective control input, a saturation function sat(.), with a modified sliding surface is proposed. Finally, Chaos in the Duffing-Holmes system with fractional orders is investigated, and a numerical simulation (synchronizing fractional-order Duffing-Holmes _ Duffing-Holmes system) are presented to show the effectiveness of the proposed controller.

Motivation & Objective

• To address the challenge of synchronizing chaotic fractional-order systems under parametric uncertainty.
• To design a robust control law that ensures finite-time synchronization despite system disturbances and model uncertainties.
• To improve synchronization performance by introducing a modified sliding surface and a saturation function to reduce chattering.
• To validate the proposed control strategy through numerical simulation of the fractional-order Duffing-Holmes system.

Proposed method

• A modified sliding surface is designed to enhance the stability and convergence properties of the synchronization error dynamics.
• A saturation function is incorporated into the control law to mitigate chattering, a common issue in conventional sliding mode control.
• The control input is derived using Lyapunov stability theory to ensure asymptotic stability of the synchronization error system.
• Fractional-order derivatives are handled using the Caputo definition, enabling accurate modeling of the chaotic system’s memory effects.
• The control law is applied to a master-slave configuration of two identical fractional-order Duffing-Holmes systems.
• Numerical simulations are performed using the Adams-Bashforth-Moulton method to solve the fractional-order system equations.

Experimental results

Research questions

• RQ1 Can a sliding mode control approach effectively synchronize two uncertain fractional-order Duffing-Holmes systems despite parametric uncertainties?
• RQ2 How does the proposed modified sliding surface improve synchronization performance compared to conventional SMC methods?
• RQ3 To what extent does the saturation function reduce chattering in the control input while maintaining system stability?
• RQ4 What is the convergence speed and robustness of the synchronization under external disturbances and model uncertainties?

Key findings

• The proposed sliding mode control scheme successfully achieves synchronization of the uncertain fractional-order Duffing-Holmes systems within a finite time.
• The use of a saturation function significantly reduces chattering in the control signal, improving practical implementability.
• The modified sliding surface enhances the convergence rate and robustness of the synchronization process.
• Numerical simulations demonstrate stable and fast synchronization, confirming the theoretical analysis.
• The system exhibits robust performance under parametric uncertainties and external disturbances, validating the controller's effectiveness.