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 linear systems and a synthesis via lossless embedding of the scattering matrix. Finally, coprime factorizations for this class of systems are developed.