A Validated Integration Algorithm for Nonlinear ODEs using Taylor Models and Ellipsoidal Calculus

Authors

• B. Houska, M.E. Villanueva, B. Chachuat

Reference

• In Proceedings of the
  52nd IEEE Conference on Decision and Control,
  Florence, Italy, pages 484 - 489, 2013.

Abstract

This paper presents a novel algorithm for bounding the reachable set of parametric nonlinear differential equations. This algorithm is based on a first-discretize-then-bound approach to enclose the reachable set via propagation of a Taylor model with ellipsoidal remainder, and it accounts for truncation errors that are inherent to the discretization. In contrast to existing algorithms that proceed in two phases — an a priori enclosure phase, followed by a tightening phase — the proposed algorithm first predicts a continuous-time enclosure and then seeks a maximal step-size for which validity of the predicted enclosure can be established. It is shown that this reversed approach leads to a natural step-size control mechanism, which no longer relies on the availability of an a priori enclosure. Also described in the paper is an open-source implementation of the algorithm in ACADO Toolkit. A simple numerical case study is presented to illustrate the performance and stability of the algorithm.

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