Virtually coupled train set coordinated station stopping control based on contractive model predictive control

Virtual coupling (VC) is an emerging technology, and cooperative stopping is a critical scenario for virtually coupled train sets (VCTS) in urban rail transit. To safely and smoothly reduce inter-train spacing and minimize stopping-time differences, this paper proposes a coordinated station stopping control method for VCTS. Initially, to characterize train braking, the unit-train dynamic model incorporates running resistance and the electro-pneumatic blending phase. Subsequently, a contractive model predictive control (CMPC) framework is developed, employing a contraction constraint to guarantee uniform asymptotic stability of the closed-loop tracking system. Furthermore, to reduce the conservatism of standard CMPC near the stopping point, a switching strategy using reachable sets is introduced. Finally, numerical simulations show that the proposed method suppresses high-frequency control chattering, improves passenger comfort, and reduces stopping-time differences by 40.9%–55.6% compared with standard CMPC. Sensitivity analyses verify robustness to unmodeled low-speed friction and V2V prediction errors while maintaining safety and synchronization.