Certifiable Explicit Model Predictive Control for Spacecraft Rendezvous under Bounded Disturbances

Abstract

Two properties of optimization-based controllers such as model predictive control (MPC) limit their use in space flight. The online computing time varies and can exceed the sampling period, and the closed-loop behavior carries no formal guarantee. Explicit MPC, where a parametric solution of the optimization problem is computed, replaces the on-board optimization with a lookup table of piecewise-affine functions with a fixed execution time. However, it has been regarded as only applicable to small horizons, since the number of regions bounds the memory the table occupies and admits only an exponential bound in the horizon. This paper builds on recent developments in parametric solutions of quadratic programs and nonconvex reachability analysis to provide an entire pipeline for spacecraft rendezvous controllers under the Clohessy-Wiltshire dynamics. For this application, efficient data structures can be constructed to encode the control law, both from a computing time and a memory standpoint. In addition, the controller can be verified offline with a guaranteed closed-loop stability certificate by resorting to reachability analysis with hybrid zonotopes. At a 51-step horizon the rendezvous law occupies 10.5 MB and is evaluated in under 300 ns. The occupied memory would fit for instance the CubeSat flight computer ARM9-class AT91RM9200 meaning that the horizon limitation does not hold when the partition is built with the parametric solvers and data structures shown in this paper.

12 pages, 5 figures, 2 tables. Submitted to IEEE Transactions on Control Systems Technology