Formula omitted]: Over- and Under-approximating Reachable Sets for analytic time-invariant differential equations

Abstract

We present OURS, a precision-oriented MATLAB tool for computing Over- and Under- approximations of Reachable Sets for analytic time-invariant differential equations. The main theoretical framework behind OURS is introduced, including the concept of evolution function, whose zero sub-level sets are used to describe reachable sets, and a series representation of evolution function. Especially, using the partial sums of this series, OURS finds over- and under-approximations of evolution function at time-instants: it consecutively estimates each remainder of the corresponding partial sum of the series with interval arithmetics until one remainder satisfies the designated precision, and then builds over- and under-approximations with this remainder. The structure of OURS is also presented, such as the forms of inputs and outputs, and technical implementations of the crucial steps inside. Moreover, we compare OURS with two other existing methods on some benchmarks. Finally, OURS is additionally extended to deal with a class of time-invariant differential systems with disturbances described by uncertain parameters. The performance of OURS in dealing with parametric uncertainties is also shown by two examples with comparisons and discussions.