Energy concepts in the state-space theory of nonlinear n-ports: Part I-Passivity

This paper is the first in a two-part sequence which aims to state rigorously the energy-based concepts which are fundamental to nonlinear network theory, passivity and losslessness, and to clarify the way they enter the input-output and the state-space versions of the subject. In this part we examine the conflicting definitions of passivity which exist in the literature and demonstrate the contradictions between them with several examples. We propose a particular definition of passivity which avoids these contradictions by eliminating the dependency on a state of "zero stored energy," and we show that it has the appropriate properties of representation independence and closure. We apply it to several specific classes of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -ports and derive equivalent passivity criteria. The exact conditions are given under which this definition is equivalent to one based on an internal energy function, and we use the concept of an internal energy function to provide a canonical network realization for a class of passive systems.