Stabilization of slow-fast control systems: the non-hyperbolic case

TL;DR

This paper addresses the stabilization of slow-fast control systems with non-hyperbolic points by employing geometric desingularization and multiple charts to improve local stability and region of attraction.

Contribution

It introduces a novel approach using multiple charts in geometric desingularization to stabilize non-hyperbolic points in slow-fast systems, expanding previous methods.

Findings

Controllers successfully stabilize non-hyperbolic points.

Multiple charts enhance the region of attraction.

Numerical examples validate the approach.

Abstract

In this paper we study the stabilization problem of a general class of slow-fast systems with one fast and arbitrarily many slow states. Moreover, the class of systems under study is slowly actuated, meaning that only the slow states are subject to the action of a controller. Furthermore, we are particularly interested in the case where normal hyperbolicity is lost. We show that by using the Geometric Desingularization method, it is possible to design controllers to locally stabilize non-hyperbolic points of any finite degeneracy. The main novelty of this paper is that, unlike previous research on the topic, we make use of more than one chart of the blow up space to enhance the region of attraction of the operating point. A couple of numerical examples highlight our contribution.