ESO-Enhanced Actor–Critic Reinforcement Learning-Optimised Trajectory Tracking Control for 3-DOF Marine Vessels

This paper develops an extended-state-observer (ESO)-enhanced actor–critic reinforcement learning (RL) scheme for the trajectory tracking control of 3-DOF marine vessels subject to uncertain hydrodynamics and environmental disturbances. A coordinate-consistent error construction is provided to obtain an exact strict-feedback second-order uncertain template. On this basis, an Hamilton–Jacobi–Bellman (HJB)-inspired optimised control structure is implemented: the critic approximates the optimal value-gradient and the actor generates the optimised control law. A key simplification is employed: rather than minimising the squared Bellman residual via complex gradients, we introduce an HJB-inspired actor–critic consistency regularisation through a weight-matching coupling. This yields computationally light online update laws and enables transparent Lyapunov-based stability analysis while not claiming exact HJB satisfaction or policy optimality. The ESO estimates lumped uncertainty and provides feedforward compensation, so the RL module learns only the observer residual. A composite Lyapunov analysis establishes the semi-global uniform ultimate boundedness of tracking errors and boundedness of all observer signals. Practical implementation with thruster allocation, explicit wind–wave–current disturbance shaping filters, and a theory-aligned ablation protocol are provided for reproducibility.