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

Abstract

A dual relation between port-Hamiltonian systems and the Brayton-Moser equations for nonlinear switched RLC circuits Jeltsema, Dimitri; Scherpen, Jacquelien M.A. CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. AbstractIn the last decades, several researchers have concentrated on the dynamic modeling of nonlinear electrical circuits from an energy-based perspective. A recent perspective is based on the concept of port-Hamiltonian (PH) systems. In this paper, we discuss the relations between the classical Brayton-Moser (BM) equations-stemming from the early sixties-and PH models for topologically complete nonlinear RLC circuits, with and without controllable switches. It will be shown that PH systems precisely dualize the BM equations, leading to possible advantages at the level of controller design. Consequently, useful and important properties of the one framework can be translated to the other. Control designs for the PH model cannot be directly implemented since they require observation of ux and charges, which are not directly available through standard sensors, while the BM models require only observation of currents and voltages. The introduced duality allows to pull back PH designs to the space of currents and voltages. This o ers the possibility to exchange several di erent techniques, available in the literature, for modeling, analysis and controller design for RLC circuits. Illustrative examples are provided to emphasize the duality between both frameworks. ?

Proportional Plus Integral Control for Set-Point Regulation of a Class of Nonlinear RLC Circuits

“…so, by noting that J , h 11 and h 22 are skew symmetric, we verify thaṫ 15) which shows that the rate at which the stored energy increases equals the difference between the power delivered by the sources and the power dissipated by the resistors.…”

“…Note, however, that in the general nonlinear case, practical use of these procedures is impeded by the requirement of an explicit solution of (2.10). An alternative way to relax Assumption 2.3 is to enforce Kirchhoff's laws using the notion of port-Hamiltonian models with constraints [15].…”

Tuning of Passivity-Preserving Controllers for Switched-Mode Power Converters

“…This property is of main importance in the following section. Other similarities and dualities between the BM equations and port-Hamiltonian systems can be found in [4]. …”

“…Remark 1: In contrast to the Lagrangian or Hamiltonian functions, that are usually defined by the circuit's total (co-)energy, the mixed-potential consists of terms related to the power circulating in the circuit. However, it is easily seen that the circuit's total co-energy, denoted by , forms a fundamental part of the BM equations, e.g., [4], i.e., we may replace and in (1) by respectively. This property is of main importance in the following section.…”

“…Again, we point out that the main reason for studying these two converters is that they describe in form and function a large family of power converter structures. A complicating property of the boost converter, as for many more converter structures, is that due to the nonlinear behavior of the conversion ratio the converter's natural resonance frequency is varying with the desired output voltage 4 . This means that the tuning criterion will also depend on the desired output voltage.…”