  We study the problem of computing robust controllable sets for discrete-time
linear systems with additive uncertainty. We propose a tractable and scalable
approach to inner- and outer-approximate robust controllable sets using
constrained zonotopes, when the additive uncertainty set is a symmetric,
convex, and compact set. Our least-squares-based approach uses novel
closed-form approximations of the Pontryagin difference between a constrained
zonotopic minuend and a symmetric, convex, and compact subtrahend. Unlike
existing approaches, our approach does not rely on convex optimization solvers,
and is projection-free for ellipsoidal and zonotopic uncertainty sets. We also
propose a least-squares-based approach to compute a convex, polyhedral
outer-approximation to constrained zonotopes, and characterize sufficient
conditions under which all these approximations are exact. We demonstrate the
computational efficiency and scalability of our approach in several case
studies, including the design of abort-safe rendezvous trajectories for a
spacecraft in near-rectilinear halo orbit under uncertainty. Our approach can
inner-approximate a 20-step robust controllable set for a 100-dimensional
linear system in under 15 seconds on a standard computer.
