Geometric Properties of Dynamic Nonlinear Networks: Transversality, Local-Solvability and eventual Passivity

Abstract

This paper gives several basic results on dynamic nonlinear networks from a geometric point of view. One of the main advantages of a geometric approach is that it is coordinate-free, i.e., results obtained by a geometric method do not depend on the particular choices of a tree, a loop matrix, state variables, etc. Therefore, the method is suitable for studying intrinsic properties of networks. It is shown that transversality of resistor constitutive relations and Kirchhoff space is a sufficient condition for the configuration space to be a submanifold. Main result of the paper states that a network is locally solvable, i.e., the dynamics of a network is well defined in the sense of Definition 3, if and only if, capacitor charges and inductor fluxes serve as a local coordinate system for the configuration space. In other words, if all the variables in a network are determined as functions of capacitor charges and inductor fluxes, at least locally, then the dynamics is well defined. Conversely, if the dynamics is well defined, then all the variables in a network are determined as functions of capacitor charges and inductor fluxes. Because of its coordinate-free property, the main result also says that if the dynamics is well defined in terms of some coordinate system, then it must be well defined in terms of capacitor charges and inductor fluxes. Conversely, if the dynamics is not well-defined in terms of capacitor charges and inductor fluxes, then there is no choice of variables in terms of which the dynamics is well defined in the sense of Definition 3. This shows that capacitor charges and inductor fluxes are the fundamental quantities in describing the dynamics of networks. Perturbation results are given which guarantee transversality and local solvability. Finally, several other perturbation results are given which guarantee eventual strict passivity of dynamic nonlinear networks. They explain why the voltage and current waveforms of almost every network of practical importance are eventually uniformly bounded.