Energy methods for lossless systems using quadratic differential forms

In this thesis, we study the properties of lossless systems using the concept of quadratic differential forms (QDFs). Based on observation of physical linear lossless systems, we define a lossless system as one for which there exists a QDF known as an energy function that is positive along nonzero trajectories of the system and whose derivative along the trajectories of the system is zero if inputs to the system are made equal to zero. Using this deffnition, we prove that if a lossless system is autonomous, then it is oscillatory. We also give an algorithm whose output is a two-variable polynomial that induces anenergy function of a lossless system and we describe a suitable way of splitting a given energy function into its potential and kinetic energy components. We further study the space of QDFs for an autonomous linear lossless system, and note that this space can be decomposed into the spaces of conserved and zero-mean quantities. We then show that there is a link between zero-mean quantities and generalized Lagrangians of an autonomous linear lossless system.Finally, we study various methods of synthesis of lossless electric networks like Cauer and Foster methods, and come up with an abstract deffnition of synthesis of a positive QDF that represents the total energy of the network to be synthesized. We show that Cauer and Foster method of synthesis can be cast in the framework of our definition. We show that our definition has applications in stability tests for linear systems, and we also give a new Routh-Hurwitz like stability test