Controlled Lagrangians and Stabilization of Discrete Mechanical Systems I

TL;DR

This paper extends controlled Lagrangian methods to discrete mechanical systems, addressing stabilization with symmetry considerations and introducing new phenomena and techniques specific to the discrete context.

Contribution

It develops a discrete controlled Lagrangian framework, including new matching conditions and parameters, for stabilizing equilibria and relative equilibria in discrete systems.

Findings

New phenomena in discrete controlled Lagrangian theory.

Introduction of momentum level selection and new parameters for kinetic matching.

Application to cart-pendulum stabilization on an incline.

Abstract

Controlled Lagrangian and matching techniques are developed for the stabilization of relative equilibria and equilibria of discrete mechanical systems with symmetry as well as broken symmetry. Interesting new phenomena arise in the controlled Lagrangian approach in the discrete context that are not present in the continuous theory. In particular, to make the discrete theory effective, one can make an appropriate selection of momentum levels or, alternatively, introduce a new parameter into the controlled Lagrangian to complete the kinetic matching procedure. Specifically, new terms in the controlled shape equation that are necessary for potential matching in the discrete setting are introduced. The theory is illustrated with the problem of stabilization of the cart-pendulum system on an incline. The paper also discusses digital and model predictive controllers.