On Excessive Transverse Coordinates for Orbital Stabilization of Periodic Motions

TL;DR

This paper investigates the use of transverse coordinates for stabilizing periodic motions in nonlinear control systems, highlighting limitations of excessive coordinates and proposing a comparison system for improved control design.

Contribution

It introduces a linearization framework for transverse coordinates, analyzes the limitations of excessive coordinates, and proposes a comparison system approach for stabilizing periodic motions.

Findings

Linearization of transverse dynamics is explicitly derived.

Excessive coordinates have limitations for control design.

A comparison system approach improves stability control.

Abstract

This paper explores transverse coordinates for the purpose of orbitally stabilizing periodic motions of nonlinear, control-affine dynamical systems. It is shown that the dynamics of any (minimal or excessive) set of transverse coordinates, which are defined in terms of a particular parameterization of the motion and a strictly state-dependent projection operator recovering the parameterizing variable, admits a (transverse) linearization along the target motion, with explicit expressions stated. Special focus is then placed on a generic excessive set of orthogonal coordinates, revealing a certain limitation of the "excessive" transverse linearization for the purpose of control design. To overcome this limitation, a linear comparison system is introduced, and conditions are stated for when the asymptotic stability of its origin corresponds to the asymptotic stability of the origin of linearized transverse dynamics. This allows for the construction of feedback controllers utilizing this comparison system which, when implemented on the dynamical system, renders the desired motion asymptotically stable in the orbital sense.