An extension to the theory of controlled Lagrangians using the Helmholtz conditions

TL;DR

This paper extends the theory of controlled Lagrangians by utilizing Helmholtz conditions to derive and recover matching conditions, enabling configuration-only feedback controls for certain mechanical systems, demonstrated on inverted pendulums.

Contribution

The paper introduces a novel approach using Helmholtz conditions to derive new matching conditions for controlled Lagrangians, simplifying control design for specific mechanical systems.

Findings

Recovered existing matching conditions using Helmholtz criteria

Derived new matching conditions for systems with configuration-only feedback

Validated the approach on inverted pendulum systems

Abstract

The Helmholtz conditions are necessary and sufficient conditions for a system of second order differential equations to be variational, that is, equivalent to a system of Euler-Lagrange equations for a regular Lagrangian. On the other hand, matching conditions are sufficient conditions for a class of controlled systems to be variational for a Lagrangian function of a prescribed type, known as the controlled Lagrangian. Using the Helmholtz conditions we are able to recover the matching conditions from [8]. Furthermore we can derive new matching conditions for a particular class of mechanical systems. It turns out that for this class of systems we obtain feedback controls that only depend on the configuration variables. We test this new strategy for the inverted pendulum on a cart and for the inverted pendulum on an incline.