Robustly Forward Invariant Sets for Mixed-Monotone Systems

Safety for dynamical systems is often posed as an invariance constraint, requiring the system trajectory to remain in some safe subset of the state-space for all time. This note presents new tools for studying reachability and set invariance for nondeterministic systems subject to a disturbance input using the theory of mixed-monotone dynamical systems. The vector field of a mixed-monotone system is characterized as being decomposable into increasing and decreasing components, which allows the dynamics to be embedded in a higher dimensional embedding system. Even though the original system is nondeterministic due to the unknown disturbance input, the embedding system has no disturbance and a single simulation of the embedding system provides bounds for reachable sets of the original dynamics. In this article, we present an efficient method for identifying robustly forward invariant and attractive sets for mixed-monotone systems by studying equilibria and their stability properties of the corresponding embedding system. We show how this approach can be applied to either the backward-time dynamics or a set of linearly transformed dynamics to establish different robustly forward invariant sets for the original dynamics, and we show also how periodic solutions to the embedding system establish invariant regions for the original dynamics as well. The findings of this work are demonstrated through two numerical examples and two case studies, including a five-dimensional planar quadrotor system.