Interval Reachability Analysis using Second-Order Sensitivity

TL;DR

This paper introduces a novel interval reachability method for nonlinear systems using second-order sensitivity matrices, providing guaranteed over-approximations with adjustable precision.

Contribution

It presents a new approach leveraging second-order sensitivities and interval arithmetic to improve over-approximation accuracy in reachability analysis.

Findings

Provides guaranteed over-approximations of the reachable set.

Achieves arbitrary precision by increasing sampling.

Extends existing methods relying only on first-order sensitivities.

Abstract

We propose a new approach to compute an interval over-approximation of the finite time reachable set for a large class of nonlinear systems. This approach relies on the notions of sensitivity matrices, which are the partial derivatives representing the variations of the system trajectories in response to variations of the initial states. Using interval arithmetics, we first over-approximate the possible values of the second-order sensitivity at the final time of the reachability problem. Then we exploit these bounds and the evaluation of the first-order sensitivity matrices at a few sampled initial states to obtain an over-approximation of the first-order sensitivity, which is in turn used to over-approximate the reachable set of the initial system. Unlike existing methods relying only on the first-order sensitivity matrix, this new approach provides guaranteed over-approximations of the first-order sensitivity and can also provide such over-approximations with an arbitrary precision by increasing the number of samples.