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.

Characterizing Inductive and Capacitive Nonlinear RLC Circuits: A Passivity Test

Linear time-invariant RLC circuits are said to be inductive (capacitive) if the current waveform in sinusoidal steady-state has a negative (resp., positive) phase shift with respect to the voltage. Furthermore, it is known that the circuit is inductive (capacitive) if and only if the magnetic energy stored in the inductors dominates (resp., is dominated by) the electrical energy stored in the capacitors. In this paper we propose a framework, based on passivity theory, that allows to extend these intuitive notions to nonlinear RLC circuits.

A Reactive Port-Hamiltonian Circuit Description and Its Control Implications

This paper first addresses the question when a given (possibly nonlinear) RGLC circuit can be rewritten as a port-Hamiltonian (PH) system—with state variables the inductor currents and capacitor voltages instead of the fluxes and charges, respectively. The question has an affirmative answer for a class of circuits that fulfills a certain regularity condition. This class includes circuits where all dynamic elements are linear, and the associated resistors and conductors are passive—though possibly nonlinear. Interestingly, the resulting Hamiltonian function is related with the circuits instantaneous reactive power associated with the inductors and capacitors. This novel circuit representation, called a reactive port-Hamiltonian description, naturally suggests a new set of non–standard passive outputs, which are shown to be useful for the design of reactive power compensation schemes. A Van der Pol oscillator circuit is used to illustrate the developments throughout the paper.