On the Existence of Lagrangians for Clarke and Park Transformed Switched-Mode Electrical Networks

Abstract

It is well-known that the classical passivity-based control (PBC) design is founded on the existence of the Euler-Lagrange (EL) equations. For a large class of systems, including mechanical, electrical, and electro-mechanical systems, its application is usually rather straightforward and requires a standard Lagrangian formulation. For three-phase electrical networks, with or without switches, it is often desirable to reformulate the resulting equations of motion into an orthogonal reference frame using the Clarke or Park transformation, or a combination of both. However, such transformations are always performed after the EL equations have been set up and a formal procedure to obtain the transformed equations directly from a Lagrangian thus far lacks in the literature. The contribution of this paper is to show that, by invoking the Helmholtz conditions and the concept of the traditor, such Lagrangians do exist, but require the inclusion of certain non-energetic cross terms.