Controllability of linear passive network behaviors

Abstract

Classical RLC realization procedures (e.g. Bott–Duffin) result in networks with uncontrollable driving-point behaviors. With this motivation, we use the behavioral framework of Jan Willems to provide a rigorous analysis of RLC networks and passive behaviors. We show that the driving-point behavior of a general RLC network is stabilizable, and controllable if the network contains only two types of elements. In contrast, we show that the full behavior of the RLC network need not be stabilizable, but is marginally stabilizable. These results allow us to formalize the phasor approach to RLC networks using the notion of sinusoidal trajectories, and to address an assumption of conventional phasor analysis. Finally, we show that any passive behavior with a hybrid representation is stabilizable. This paper relies substantially on the fundamental work of our late friend and colleague Jan Willems to whom the paper is dedicated.