Constrained Control for Systems on Lie Groups with Uncertainties via Tube-Based Model Predictive Control on Euclidean Spaces

In this paper, the constrained control of systems evolving on matrix Lie groups with uncertainties is considered. The proposed methodology is composed of a nominal Model Predictive Control (MPC), and a feedback controller. The previous work on the control of systems on manifolds is applied to design the nominal MPC, which generates the nominal trajectory. In the nominal MPC, the state and input constraints on the Lie group are transformed into the constraints on the Euclidean space. While to deal with uncertainties, the feedback control used to track the nominal trajectory is designed directly on the Lie group. The tracking error in the feedback control is proved to be bounded in invariant sets. Such invariant sets are further used to revise the constraints in nominal MPC. We prove that by using this methodology, the stability and safety of the system can be guaranteed simultaneously. The proposed methodology is applied to the constrained attitude control of rigid bodies. In the application example, the detailed mathematical proof and the numerical simulation are presented, illustrating the feasibility of the proposed methodology.