Document Type : Research Article
Authors
Department of Mathematics, Payame Noor University (PNU), Iran
10.30473/coam.2025.74158.1299
Abstract
This paper introduces a robust hybrid adaptive control framework for stabilizing chaotic systems under persistent, potentially large time delays. The controller is based on an enhanced Lyapunov–Krasovskii functional that integrates an energy-capturing integral term with a bounded trigonometric term. The integral term accounts for historical effects by quantifying cumulative energy over the delay period, while the trigonometric term attenuates nonlinear oscillations. Embedding these components in a single control law yields stabilization of all state variables to the equilibrium despite substantial delays. We establish Uniform Ultimate Boundedness, showing that trajectories enter a compact neighborhood of the equilibrium after a finite transient and subsequently converge. Adjustable gains enable practitioners to determine the convergence radius and the size of the attraction region according to practical requirements. The method is validated on the delayed Lorenz system; simulations with a 20-second delay demonstrate rapid convergence to a small neighborhood of the equilibrium, with the Lyapunov functional derivative remaining non-positive. A comparative study with established controllers underscores the proposed approach’s favorable trade-offs among computational cost, oscillation suppression, and explicit stability guarantees. Overall, the proposed framework delivers a practical, robust, and high-performance solution for controlling chaotic systems in the presence of large time delays.
Highlights
- Introduces an enhanced Lyapunov function and a novel hybrid adaptive controller within the Lyapunov–Krasovskii framework, combining: (i) an integral term to mitigate delay effects and (ii) a bounded trigonometric term to limit nonlinear oscillations and boost stability.
- Provides a rigorous mathematical analysis establishing positive definiteness of the Lyapunov function and negative definiteness of its derivative, guaranteeing global convergence of state variables to the equilibrium under constant delays.
- Demonstrates controller performance via simulations of the chaotic Lorenz system under both small and large delays (up to 20 seconds), showing a uniform reduction in system energy through outputs such as phase portraits and Lyapunov derivative plots.
- Addresses delay- and nonlinearity-induced challenges by proposing mechanisms to suppress adverse effects and prevent oscillation amplification.
- Illustrates practical applicability in real-world systems with processing or communication delays across sectors like power networks, chemical reactors, and IoT.
- Acknowledges limitations and offers directions for future work to further improve efficiency and scalability.
Keywords
- Constant time delay
- Hybrid adaptive control
- Lyapunov function
- Asymptotic stability
- Practical stability
Main Subjects
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