The zero dynamics of feedback linearisation and Ehresmann connections

TL;DR

This paper explores the zero dynamics in feedback linearisation using Ehresmann connections, framing zero dynamics as motions along fibres in a fibre bundle, providing a geometric perspective on control systems.

Contribution

It introduces a novel geometric framework using Ehresmann connections to define zero dynamics as vertical vector fields on a fibre bundle.

Findings

Zero dynamics can be characterized as motions along fibres.

The fibre bundle structure provides new insights into feedback linearisation.

The formalism unifies control concepts with differential geometry.

Abstract

The notion of zero dynamics is a cornerstone of many solutions to important control problems such as feedback linearisation and disturbance decoupling. For a SISO affine control system with relative degree strictly less than the order of the system, it is known that there exists a state transformation and a feedback transformation such that the system can be transformed to a linear system. Making use of the fibre bundle structure induced by the state transformation we show that it is possible to define a connection on the state manifold such that the zero dynamics can be defined as a vertical vector field. In this formalism the zero dynamics can be understood as motions along the fibres of a fibre bundle.