Improving the Fidelity of Mixed-Monotone Reachable Set Approximations via State Transformations

Mixed-monotone systems are separable via a decomposition function into increasing and decreasing components, and this decomposition function allows for embedding the system dynamics in a higher-order monotone embedding system. Embedding the system dynamics in this way facilitates the efficient over-approximation of reachable sets with hyper-rectangles, however, unlike the monotonicity property, which can be applied to compute, e.g., the tightest hyperrectangle containing a reachable set, the application of the mixed-monotonicity property generally results in conservative reachable set approximations. In this work, we explore conservatism in the method, and we consider, in particular, embedding systems that are monotone with respect to an alternative partial order. This alternate embedding system is constructed with a decomposition function for a related system formed via a linear transformation of the initial state-space. We show how these alternate embedding systems allow for computing reachable sets with improved fidelity, i.e., reduced conservatism.