General linear nonstationary passive systems: Derivation of the explicit state model

Abstract

We consider the general nonstationary passive systems modeled by linearRCG (G = gyrator) networks. Such networks in general contain conventional topological degeneracies (i.e., capacitor-only loops and/or inductor-only cutsets) as well as the topological degeneracies due to gyrator positioning in the network. Our central result is the explict demonstration that the existence of topological degeneracies does not impose any obstruction to the existence of the explicit state model, which we derive under no restriction on the topology of the network. We also discuss the conditions for the existence of the state model, its unique solutions, and the continuity of the state vector. The nature of the degeneracies inherent in the formulation is highlighted and it is shown that a gyrator-only subnetwork is accountable for algebraic degeneracies. Since the nature and the existence of topological degeneracies does not have anything to do with whether the element characteristics are linear or nonlinear, passive or active, our results are easy to extend to a large class of nonlinear and/or activeRCG networks.