Lagrangian modeling of switching electrical networks

Abstract

In this paper, a general and systematic method is presented to model topologically complete electrical networks, with or without multiple or single switches, within the Euler-Lagrange framework. Apart from the physical insight that can be obtained in this way, the framework has proven to be useful for the application of passivity-based control techniques, which on a case by case basis already has shown to be useful for the control of power converters within the class of switching electrical networks. The switches are assumed to be ideal, and pulse-width modulation is taken into account. Magnetic coupling of inductive elements is also included in the framework.

A dual relation between port-Hamiltonian systems and the Brayton–Moser equations for nonlinear switched RLC circuits

“…A pair of magnetically coupled-inductors can be considered as special case of an inductor-only cutset. In Scherpen et al (2003) a commonly used equivalent representation is used for modeling and analysis purposes. This equivalent representation exists of three inductors forming a Y-circuit followed by an ideal one-to-one transformer to incorporate the galvanic junction.…”

“…This intermediate help variable is then eliminated using the constraint equation. With the method used in Scherpen et al (2003) the mutual inductance does not contribute a coordinate but contributes a crossterm to corresponding magnetic co-energy instead.…”

“…Almost a decade later, in Kwatny, Massimo, and Bahar (1982) a generalized Lagrangian framework is proposed in which some severe limitations of the previous methods are relaxed. After this period the area became relatively quiet, until recently with the introduction of port-Hamiltonian (PH) systems (van der Schaft, 2000) and Lagrangian modeling of power electronic systems in Ortega, Lorà a, Nicklasson, and Sira-Ramà rez (1998) and Scherpen, Jeltsema, and Klaassens (2003). In the context of switched-mode systems it is shown that the dynamics correspond to systems derivable from a Lagrangian or PH point of view.…”

A Novel Variational Method for Deriving Lagrangian and Hamiltonian Models of Inductor-Capacitor Circuits

“…A canonical system of Hamiltonian equations describing the dynamics of the heavy top is then obtained by applying the Pontryagin maximum principle on manifolds. An alternative approach to the variational modeling of electric circuits has, independently of the present research, been proposed by Scherpen, Jeltsema, and Klaassens [16] and Clemente-Gallardo and Scherpen in [5]. Whereas the present paper makes use of the Pontryagin maximum principle, the approach in these papers is based on the constrained variational principles from holonomic mechanics.…”

An Electrical Interpretation of Mechanical Systems via the Pseudo-inductor in the Brayton-Moser Equations

“…Notice that (q,q) ∈ R 2r while (i ρ , v σ ) ∈ R r+s , which corresponds to the fact that mechanical systems are "nodical" (see e.g., [2]), where as electrical systems are not. In order to cope with this, the Kirchhoff current law may be needed (see e.g., [11]). However, it can also be translated into an integral version and will be represented in one of the conditions of our theorem.…”