Invariant subspace methods in linear multivariable-distributed systems and lumped-distributed network synthesis

Linear multivariable-distributed systems and synthesis problems for lumped-distributed networks are analyzed. The methods used center around the invariant subspace theory of Helson-Lax and the theory of vectorial Hardy functions. State-space and transfer function models are studied and their relations analyzed. We single out a class of systems and networks with nonrational transfer functions (scattering matrices), for which several of the well-known results for lumped systems and networks are generalized. In particular we develop the relations between singularities of transfer functions and "natural modes" of the systems, a degree theory for infinite-dimensional finear systems and a synthesis via lossless embedding of the scattering matrix. Finally coprime factorizations for this class of systems are developed. These factorizations play an essential role in the development and show that properties of Hardy functions are of fundamental importance for this class of distributed systems as properties of rational functions are for lumped systems.