Adaptive control for a class of nonlinear complex dynamical systems with uncertain complex parameters and...

Area of Science:

• Control Theory
• Nonlinear Dynamics
• Complex Systems

Background:

• Adaptive control is crucial for stabilizing dynamic systems with uncertainties.
• Complex-variable chaotic systems (CVCSs) present unique challenges due to their multi-dimensional and complex nature.
• Existing control methods often struggle with the complexities of time-dependent CVCSs and uncertain parameters.

Purpose of the Study:

• To extend adaptive control from real to complex space for n-dimensional time-dependent strict-feedback CVCSs.
• To develop a unified framework for designing adaptive complex controllers and estimating unknown complex parameters.
• To ensure asymptotic stability of CVCSs in the presence of uncertainties and perturbations.

Main Methods:

• Development of an adaptive complex scalar controller.
• Utilization of complex update laws for parameter estimation.
• Integration of Lyapunov functions with complex-valued vectors and the back-stepping technique.
• Application of Wirtinger calculus in complex space for stability analysis.

Main Results:

• A new control scheme for n-dimensional time-dependent strict-feedback CVCSs with uncertain complex parameters and perturbations.
• Sufficient criteria for the asymptotic stabilization of CVCSs derived using complex Lyapunov functions and back-stepping.
• Successful estimation of unknown complex parameters through adaptive complex update laws.
• Validation of the proposed control scheme and theoretical results via numerical simulations.

Conclusions:

• The proposed adaptive control scheme effectively extends real-space control to complex space for CVCSs.
• The developed framework provides a robust method for stabilizing and controlling complex chaotic systems with uncertainties.
• This work offers a significant advancement in the control of complex-variable chaotic and hyperchaotic systems.