Closed-form minimum horizon bound for terminal docking

Derive a closed-form minimum prediction-horizon bound for the Operation 3 economic MPC controller with the hard terminal equality, accounting for the coupled and constrained Clohessy–Wiltshire–Hill dynamics, instead of relying on the empirically identified horizon N=24.

Background

The final docking controller imposes the terminal equality x_N=x_dock and uses a horizon of N=24. The authors determined this value by sweeping the horizon downward in simulation until the quadratic program became infeasible, so the value is an empirical feasibility threshold rather than an analytically established bound.

A closed-form derivation would characterize the minimum horizon required to reach the docking state in both position and velocity while satisfying the input, along-track, and line-of-sight constraints of the terminal approach.