Barrier-Function-Enabled Control for Vessel Systems under Dynamic Event-Triggered Protocols: The ADP Approach

In this paper, a constraint-aware suboptimal control scheme under adaptive dynamic programming (ADP) is investigated for nonlinear vessel systems characterized by dynamic event-triggering mechanisms (DETMs) and state constraints. A relaxed barrier function (RBF) is designed as a penalty term in the cost function to replace the inequality constraints, thus achieving constraints on the system state. By resorting to neural network (NN) approximation of the nonlinear dynamics, an optimal control framework is established via the dual method, combined with the Lagrange multipliers on the RBF. Within this framework, the Lagrangian multipliers are employed to balance the optimisation of control performance and state constraints. The convergence of value iteration algorithms is revealed through rigorously mathematical analysis. Furthermore, by using the Lyapunov stability theory, the desired observer gain is calculated via a set of matrix inequalities, and sufficient conditions for the relevant parameters and learning rates are derived such that the weight estimation errors of both the observer's NNs and the actor-critic NNs are ultimately bounded. Finally, simulations of both vessel systems and numerical examples are used to validate the effectiveness of the proposed method.