Load characterization and power conditioner synthesis using higher-order elements

It is shown that virtually all nonlinear and/or time-varying loads that generate harmonic current distortion can be characterized in terms of so-called higher-order circuit elements. The most relevant higher-order elements exploited in this paper are the memristor, meminductor, and memcapacitor. Such elements naturally arise by introducing constitutive relationships in terms of higher-order voltage and current differentials and integrals. Consequently, the power conditioner necessary to compensate for the load current distortions is synthesized similarly. The new characterization and compensation synthesis is applied to the half-wave rectifier and the controlled bridge converter.     

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 Moser's stability theorems that also includes the analysis of circuits that contain nonlinear resistors, conductors, inductors, and/or capacitors at the same time.     

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.     

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.     

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., Tellegen's theorem, and heavily rely on the seminal paper of Brayton and Moser in 1964.     

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.     

A power-based description of standard mechanical systems

This paper is concerned with the construction of a power-based modeling framework for mechanical systems. Mathematically, this is formalized by proving that every standard mechanical system (with or without dissipation) can be written as a gradient vector field with respect to an indefinite metric. The form and existence of the corresponding potential function is shown to be the mechanical analog of Brayton and Moser's mixed-potential function as originally derived for nonlinear electrical networks in the early sixties. In this way, several recently proposed analysis and control methods that use the mixed-potential function as a starting point can also be applied to mechanical systems.     

An energy-balancing perspective of interconnection and damping assignment control of nonlinear systems

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.     

Port-Hamiltonian description and analysis of the LuGre friction model

A port-Hamiltonian formulation of the LuGre friction model is presented that can be used as a building block in the physical modelling of systems with friction. Based on the dissipation structure matrix of this port-Hamiltonian LuGre model, an alternative proof can be given for the passivity conditions that are known in the literature. As a specific example, the interconnection of a mass with the port-Hamiltonian LuGre model is presented. It is shown that the lossless-interconnection structure and dissipation structure of the port-Hamiltonian LuGre model are consistent with those of this interconnection. As an additional example, the port-Hamiltonian formulation of a quarter-car system with a LuGre-based tyre model is presented.