Dynamic Global Feedback Stabilization: why do the twist?

TL;DR

This paper explores how the topology of the combined system and controller space affects the ability to achieve global stabilization, emphasizing the role of twisted versus trivial bundles.

Contribution

It introduces a topological framework for understanding dynamic feedback stabilization, highlighting the significance of bundle topology in control system design.

Findings

Dynamic feedback on twisted bundles can stabilize systems that trivial bundles cannot.

Topological conditions are necessary for global stabilization in dynamic feedback systems.

The study emphasizes the importance of bundle topology in control theory.

Abstract

We investigate global dynamic feedback stabilization from a topological viewpoint. In particular, we consider the general case of dynamic feedback systems, whereby the total space (which includes the state space of the system and of the controller) is a fibre bundle, and derive conditions on the topology of the bundle that are necessary for various notions of global stabilization to hold. This point of view highlight the importance of distinguishing trivial bundles and twisted bundles in the study of global dynamic feedback stabilization, as we show that dynamic feedback defined on a twisted bundle can stabilize systems that dynamic feedback on trivial bundles cannot.