MPC on manifolds with applications to the control of systems on matrix Lie groups

This paper presents a model predictive control scheme for systems with discrete-time dynamics evolving on configuration spaces that are smooth manifolds. The scheme exhibits similar properties to that of ordinary model predictive control applied to dynamics evolving on R^n. It also exhibits global asymptotic stability properties even in the case when there does not exist a globally stabilizing, continuous control law, implying that the model predictive control law is discontinuous. We also demonstrate that there do not exist globally stabilizing, continuous control laws for manifolds with Euler characteristic equal to 1. In particular, the case of matrix Lie groups is considered, which are manifolds whose Euler characteristic is equal to 0. An application to spacecraft attitude control is also considered in the paper, in which spacecraft attitude evolves on the matrix Lie group SO(3). Two simulation results are reported. The first demonstrates that the scheme is able to enforce constraints. The second simulation considers the unconstrained case in order to test properties relating to global stability and it is shown that the stabilizing model predictive control law is discontinuous.