Sampled-data reachability analysis using sensitivity and mixed-monotonicity

TL;DR

This paper introduces a novel method for over-approximating the reachable sets of uncertain continuous-time systems using sensitivity analysis and mixed-monotonicity, with scalable bounds and simulation-based estimation.

Contribution

It establishes the equivalence between sensitivity-based and mixed-monotonicity over-approximations and proposes a new scalable method applicable to any bounded sensitivity system.

Findings

The new method scales linearly with state dimension.

Simulation-based bounds estimation is effective.

Numerical examples demonstrate practical applicability.

Abstract

This paper over-approximates the reachable sets of a continuous-time uncertain system using the sensitivity of its trajectories with respect to initial conditions and uncertain parameters. We first prove the equivalence between an existing over-approximation result based on the sign-stability of the sensitivity matrices and a discrete-time approach relying on a mixed-monotonicity property. We then present a new over-approximation result which scales at worst linearly with the state dimension and is applicable to any continuous-time system with bounded sensitivity. Finally, we provide a simulation-based approach to estimate these bounds through sampling and falsification. The results are illustrated with numerical examples on traffic networks and satellite orbits.