Maximal Ellipsoid Method for Guaranteed Reachability of Unknown Fully Actuated Systems

In the face of an adverse event, autonomous systems may undergo abrupt changes in their dynamics. In such an event, systems should be able to determine their continuing capabilities to then execute a provably completable task. This paper focuses on the scenario of a change in the system dynamics following an adverse event, aiming to determine the system's guaranteed performance capabilities by finding a set of states that are provably reachable by the system. While it is obviously impossible to exactly determine the reachable set without full knowledge of the system dynamics, we present a method of determining its under-approximation while assuming only partial knowledge of the system structure. Our technical approach relies on showing that an intersection of infinitely many ellipsoids - available velocity sets for each system consistent with the partial knowledge of the dynamics - is the same as an intersection of some finite collection of ellipsoids. This result enables us to find a maximal ellipsoid lying in such an intersection, yielding a set of velocities that the system is provably able to pursue regardless of its exact dynamics.