Jacquelien M.A. Scherpen

Power shaping: a new paradigm for stabilization of nonlinear RLC circuits

It is well known that arbitrary interconnections of passive (possibly nonlinear) resistors, inductors, and capacitors define passive systems, with port variables the external source voltages and currents, and storage function the total stored energy. In this note, we prove that for a class of RLC circuits with convex energy function and weak electromagnetic coupling it is possible to add a differentiation to the port terminals preserving passivity - with a new storage function that is directly related to the circuit power. The result is of interest in circuits theory, but also has applications in control as it suggests the paradigm of power shaping stabilization as an alternative to the well-known method of energy shaping. We show in this note that, in contrast with energy shaping designs, power shaping is not restricted to systems without pervasive dissipation and naturally allows to add derivative actions in the control. These important features, that stymie the applicability of energy shaping control, make power shaping very practically appealing. To establish our results we exploit the geometric property that voltages and currents in RLC circuits live in orthogonal spaces, i.e., Tellegens theorem, and heavily rely on the seminal paper of Brayton and Moser in 1964.

Tuning of passivity-preserving controllers for switched-mode power converters

Nonlinear passivity-based control (PBC) algorithms for power converters have proved to be an interesting alternative to other, mostly linear, control techniques. The control objective is usually achieved through an energy reshaping process and by injecting damping to modify the dissipation structure of the system. However, a key question that arises during the implementation of the controller is how to tune the various control parameters. From a circuit theoretic perspective, a PBC forces the closed-loop dynamics to behave as if there are artificial resistors-the control parameters-connected in series or in parallel to the real circuit elements. In this paper, a solution to the tuning problem is proposed that uses the classical Brayton-Moser equations. The method is based on the study of a certain mixed-potential function which results in quantitative restrictions on the control parameters. These restrictions seem to be practically relevant in terms stability, overshoot and nonoscillatory responses. The theory is exemplified using the elementary single-switch buck and boost converters.

On passivity and power-balance inequalities of nonlinear RLC circuits

Arbitrary interconnections of passive (possibly nonlinear) resistors, inductors, and capacitors define passive systems, with power port variables the external source voltages and currents, and storage function the total stored energy. In this paper, we identify a class of RLC circuits (with convex energy function and weak electromagnetic coupling), for which it is possible to add a differentiation to the port terminals preserving passivity-with a new storage function that is directly related to the circuit power. To establish our results, we exploit the geometric property that voltages and currents in RLC circuits live in orthogonal spaces, i.e., Tellegens theorem, and heavily rely on the seminal paper of Brayton and Moser (1964).

Lagrangian modeling of switching electrical networks

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

In 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 flux 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 offers the possibility to exchange several different 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.

An Energy-Balancing Perspective of Interconnection and Damping Assignment Control of Nonlinear Systems

Abstract Stabilization of nonlinear feedback passive systems is achieved assigning a storage function with a minimum at the desired equilibrium. For physical systems a natural candidate storage function is the difference between the stored and the supplied energies—leading to the so-called Energy-Balancing control, whose underlying stabilization mechanism is particularly appealing. Unfortunately, energy-balancing stabilization is stymied by the existence of pervasive dissipation, that appears in many engineering applications. To overcome the dissipation obstacle the method of Interconnection and Damping Assignment, that endows the closed-loop system with a special—port-controlled Hamiltonian—structure, has been proposed. If, as in most practical examples, the open-loop system already has this structure, and the damping is not pervasive, both methods are equivalent. In this brief note we show that the methods are also equivalent, with an alternative definition of the supplied energy, when the damping is pervasive. Instrumental for our developments is the observation that, swapping the damping terms in the classical dissipation inequality, we can establish passivity of port-controlled Hamiltonian systems with respect to some new external variables—but with the same storage function

Lagrangian modeling and control of switching networks with integrated coupled magnetics

In this paper a method is presented to build an Euler-Lagrange model for electrical networks, including switches and integrated (non-ideal) coupled-magnetics, in a structured general way. One of the advantages of emphasizing the physical structure of these systems is its functionality during the controller design stage. In a case a switching network contains coupled-inductor structures, an additional path for the energy transfer is introduced. For this reason, a basic building block is proposed that describes the dynamical behaviour of a pair of magnetically coupled-inductors. This building block is applicable to all types of switching converters, and easily predicts the existence of reduced or zero-ripple current. The switches make the dynamic models nonlinear. It is shown that under certain coupling conditions it is possible to design a globally stable controller. The approach is illustrated by means of the coupled-inductor Cuk converter with zero-output ripple, in closed loop with an adaptive passivity-based controller.

On Brayton and Moser's missing stability theorem

In the early 1960s, Brayton and Moser proved three theorems concerning the stability of nonlinear electrical circuits. The applicability of each theorem depends on three different conditions on the type of admissible nonlinearities in circuit. Roughly speaking, this means that the theorems apply to either circuits that contain purely linear resistors or conductors-combined with linear or nonlinear inductors and capacitors or to circuits that contain purely linear inductors and capacitors-combined with linear or nonlinear resistors and conductors. This brief note presents a generalization of Brayton and Mosers stability theorems that also includes the analysis of circuits that contain nonlinear resistors, conductors, inductors, and/or capacitors at the same time.

Power Shaping Control of Nonlinear Systems: A Benchmark Example

It is well known that energy balancing control is stymied by the presence of pervasive dissipation. To overcome this problem in electrical circuits, the authors recently proposed the alternative paradigm of power shaping—where, as suggested by its name, stabilization is achieved shaping a function akin to power instead of the energy function. In this paper we extend this technique to general nonlinear systems and apply it for the stabilization of the benchmark tunnel diode circuit. It is shown that, in contrast with other techniques recently reported in the literature, e.g. piecewise approximation of nonlinearities, power shaping yields a simple linear static state feedback that ensures (robust) global asymptotic stability of the desired equilibrium.

Energy control of multi-switch power supplies; an application to the three-phase buck type rectifier with input filter

The first part of the paper presents a systematic method for the modeling of power electronic networks with multiple switches using the classical Lagrangian framework. The main advantage of this method is its structured and general character compared to other methods. The power of working with Lagrangian dynamics, especially when very large networks have to be analyzed, is that Kirchhoffs voltage law is given in advance, while the interconnection structure is based on Kirchhoffs current laws only. The physical nonlinear structure of these systems is revealed and can be used for feedback controller design and stability analysis. The second part concentrates on the proposed passivity-based controller design technique based on the Lagrangian structure. Besides its physical significance, this approach appears to be an interesting alternative when dealing with the problem of input filter influences in power electronic systems. With this technique, artificial damping is injected such that the impedances of the in- and output filters can be matched. In this way power is not reflected and resonance problems, especially during the start-up and transient conditions, are minimized. A unity power factor passivity-based control algorithm for the three-phase buck type rectifier with an input filter is proposed that does not require current sensors but only measurements of the capacitor voltages.