Parametrization of rational lossless matrices with applications to linear systems theory

The study of lossless matrices is motivated by two main applications in system theory: * Lossless matrices appear as cornerstone in matrix rational L2 approximation with stability constraint, * The transfer functions of conservative systems, and in particular scattering matrices of frequency filters, are lossless. The parametrizations that are presented in this work rely on Schur analysis and interpolation theory. The main outcomes of this research are: an efficient implementation of a rational approximation algorithm; a well-understood correspondence between the tangential Schur algorithm and balanced realizations, with potential applications to model reduction; a method to built a lossless symmetric (higher size, minimal degree) extension of a contractive rational matrix. A key step in understanding the polynomial structure of scattering matrices in view of filter design applications.