Template Zonotopes for Near-Rotational Cases of Nagumo's Theorem

Given a dynamical system (i.e., the equations of motions of a cyber-physical system that describe how it can move through space), an invariant set is a region of space with the property that any system that starts within that set, remains in that region forever. This property is very useful when trying to stabilize autonomous systems, e.g., if an autonomous vehicle needs to follow a trajectory despite disturbances arising from the wind etc. In this context, the Nagumo theorem is a famous result that provides an equivalent characterization of invariant sets, which in certain situations can be expressed as an optimization problem that can be solved numerically. This is the case for ellipsoids and V-polytopes, but recent advances have extended this idea to zonotopes, with some limitations.

In particular, the resulting procedure for zonotopes relies on the choice of a template matrix, which selects the main directions of the zonotope. This template, if chosen in a clever manner, can ensure that the set is as large as possible. The purpose of this study is to find new heuristics on how to choose this template in the case of near-rotational dynamics (i.e., in the case where trajectories of the system are almost circular).

This topic involves a combination of optimization theory and linear algebra, and will offer a short introduction to deductive verification. Applicants should have some basic knowledge about ordinary differential equations (ODEs) and some very basic knowledge about MATLAB programming.

For more information, please reach out to adrian.kulmburg@kit.edu.