Stabilization of slow-fast systems at fold points

TL;DR

This paper presents a novel geometric desingularization method to stabilize slow-fast control systems at fold points, overcoming limitations of classical singular perturbation theory, with application to an electric circuit example.

Contribution

It introduces a new stabilization approach for non-hyperbolic fold points in slow-fast systems using geometric desingularization techniques.

Findings

Successful stabilization of a fold point demonstrated on an electric circuit

New geometric desingularization method effective for non-hyperbolic singularities

Overcomes limitations of classical singular perturbation theory in control design

Abstract

In this document, we deal with the stabilization problem of slow-fast systems (or singularly perturbed Ordinary Differential Equations) at a non-hyperbolic point. The class of systems studied here have the following properties: 1) they have one fast variable and an arbitrary number of slow variables, 2) they have a non-hyperbolic singularity of the fold type at the origin. The presence of the aforementioned singularity complicates the analysis and the controller design of such systems. In particular, the classical theory of singular perturbations cannot be used. We show a novel design process based on geometric desingularization, which allows the stabilization of a fold point of singularly perturbed control systems. Our results are exemplified on an electric circuit.