  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 hyperrectangles, 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, 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.
