Analytical solutions of the controller matching problem based on inverse lqr for spacecraft relative dynamics

This paper studies analytical solutions of the controller matching problem based on inverse LQR which works for any prediction horizon. The solutions are applied to model predictive control (MPC) to make its performance the same as a preassigned linear state-feedback controller when constraints are not active. The problem is solved for linearized relative dynamics of spacecraft described by the Clohessy-Wiltshire equations and the solutions comprise the weighting matrices associated with the MPC optimization problem. The designed MPC controller will be useful in the application of spacecraft rendezvous and docking as it behaves as the predesigned linear controller when constraints are inactive, while satisfying the constraints in an optimal manner when they are active. The solutions are separately derived for each of the out-of-plane and in-plane cases. Numerical simulations are carried out to demonstrate the effectiveness of the analytical solutions proposed herein.