Adaptive Neural Tracking Control of a Class of Hyperbolic PDE With Uncertain Actuator Dynamics

Area of Science:

• Control Systems Engineering
• Applied Mathematics
• Neural Networks

Background:

• Adaptive control is crucial for systems with unknown dynamics.
• Hyperbolic partial differential equations (PDEs) with boundary actuators present significant control challenges.
• Existing research primarily addresses stabilization, not tracking, for these systems.

Purpose of the Study:

• To develop an adaptive neural tracking control strategy for hyperbolic PDEs with nonlinear ordinary differential equation (ODE) boundary dynamics.
• To address the challenge of unknown nonlinearities in the ODE subsystem.
• To advance the field by focusing on tracking control, differentiating from prior stabilization-focused work.

Main Methods:

• Formulation of a virtual exosystem to generate reference trajectories.
• Design of a novel adaptive geometric controller utilizing neural networks (NNs) for nonlinearity estimation.
• Application of finite and infinite-dimensional backstepping techniques.
• Lyapunov theory for rigorous stability analysis of the closed-loop system.

Main Results:

• Successful implementation of adaptive neural tracking control for the specified PDE system.
• Effective estimation and compensation of unknown nonlinearities via NNs.
• Demonstrated stability of the closed-loop system through theoretical proofs.
• Validation of the control strategy via two numerical simulations.

Conclusions:

• The proposed adaptive geometric controller effectively achieves tracking control for hyperbolic PDEs with nonlinear ODE boundary dynamics.
• Neural networks provide a viable approach for estimating and compensating unknown system nonlinearities.
• The developed control framework offers a significant advancement over existing stabilization-only methods.