Optimal Polynomial Control for Finite Horizon Problems in Virtual Train Coupling

Abstract

Virtual Coupled Train Sets (VCTS) is an emerging railway technology that aims to enhance rail transport efficiency by allowing trains to operate at reduced inter-vehicular distances, akin to vehicle platooning in road transport. This thesis focuses on developing optimal polynomial control strategies for finite horizon problems in the context of VCTS. Specifically, we propose a mathematical framework for train-following control based on value function approximation techniques. We begin by formulating a nonlinear third-order vehicle dynamic model incorporating traction force delay and resistive forces using the Davis equation. Existence and uniqueness properties of system solutions are established through Carathéodory's theorem, and we conduct a phase diagram analysis to explore system stability and equilibrium behaviors. The optimal control problem is then derived using the Hamilton-Jacobi-Bellman (HJB) framework, leading to a feedback law for real-time train operation. To address the computational challenges of solving the HJB equation, we employ a learning-based approach. We approximate the value function using polynomial basis functions and optimize its parameters via an adjoint-based gradient computation technique. The proposed method is validated through numerical simulations, demonstrating its effectiveness in optimizing train trajectories while ensuring safety constraints. The results highlight the potential of polynomial-based optimal control for VCTS applications, providing a scalable and interpretable alternative to conventional control methods. Future work may explore extensions incorporating stochastic disturbances and real-time implementation in practical railway scenarios.