Balanced realization of lossless systems: Schur parameters, canonical forms and applications

Lossless systems have many applications in systems and control theory, including signal processing, filter design, system identification, system approximation, and the parameterization of classes of linear systems. In this survey paper we address the issue of parameterization of the space of rational lossless matrix functions by successfully combining two approaches. The first approach proceeds in state-space from balanced realizations and triangular pivot structures of reachability matrices. The second approach concerns interpolation theory with linear fractional transformations and the tangential Schur algorithm. We construct balanced realizations (and canonical forms) in terms of the Schur parameters encountered in the tangential Schur algorithm, and conversely, we interpret balanced realizations in discrete-time and in continuous-time in terms of Schur parameters. A number of application areas are discussed to illustrate the importance of this theory for a variety of topics.