## Section: New Results

### Interpolation and parametrizations of transfer functions

Participants : Martine Olivi, Bernard Hanzon, Ralf Peeters [ Univ. Maastricht ] .

Our work of the past ten years on balanced realizations of lossless systems, Schur parameters, canonical forms and applications were the topic of a semi-plenary session, given by R. Peeters at the conference Sysid09 [28] . Our last results on subdiagonal pivot structure for input-normal pairs and associated canonical forms were also presented at this conference [27] . These forms generalise to the MIMO case the well-known Hessenberg form in discrete-time and Schwarz-Ober form in continuous-time. Their use for model reduction purposes seems to be relevant and is currently under investigation.

For the class of lossless discrete-time systems, subdiagonal forms can be computed from a specific backward recursive Schur algorithm. In continuous-time, the relevant recursive algorithm in connection with these forms involves a boundary interpolation problem. We got a parametrization of the (subdiagonal) Ober-Schwarz canonical form (SISO) in terms of boundary interpolation values (angular derivatives). These results were presented at the ERNSI meeting (poster). Boundary interpolation of matrix lossless functions and its applications to the parametrization of filter banks leading to orthogonal wavelets is under study (see section 6.5 ).