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

TL;DR

This paper introduces a novel approach to reduce conservatism in mixed-monotone reachable set approximations by employing state transformations and alternative partial orders, leading to more accurate over-approximations.

Contribution

It proposes a method using linear state transformations to construct alternative embedding systems, improving the fidelity of reachable set approximations in mixed-monotone systems.

Findings

Reduced conservatism in reachable set approximations.

Enhanced accuracy through alternative partial order embeddings.

Theoretical validation of improved fidelity.

Abstract

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.