Energy Concepts in the State-Space Theory of Nonlinear n-Ports: Part II - Losslessness

This paper is the second in a two-part series [1] that aims to provide a rigorous foundation in the nonlinear domain for the two energybased concepts fundamental to network theory: passivity and losslessness. We hope to clarify the way they enter into both the state-space and the input-output viewpoints. Our definition of losslessness is modeled on that of a "conservative system" in classical mechanics; several examples are used to compare it with other concepts of losslessness currently found in the literature. We show in detail how this definition avoids the anomalies and contradictions that many other definitions produce. This concept of losslessness has the desirable property of being preserved under interconnections, and we extend it to one that is representation independent as well. It is applied to five common classes of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -ports, yielding explicit criteria for losslessness in terms of the state and output equations. In particular we give a rigorous justification for the various equivalent criteria in the linear case. A network realization and a new explicit canonical form of state representation are derived for a large class of lossless nonlinear systems.