This talk gives an overview of the methods and results described in my thesis. The main topic of the thesis is modeling a class of linear time-invariant systems. The system is parameterized in the context of the interpolation theory with a degree constraint. In the thesis, this parameterization is the key tool to design models of dynamical systems and the optimization theory is the major tool for parameter estimation of these models.
In the first part of the talk, we formulate two related interpolation problems with a degree constraint, the analytic interpolation theory with a degree constraint and the rational covariance extension theory, and the corresponding fundamental results developed by Byrnes, Lindquist, Georgiou and coauthors in the last decades.
The second part of the talk is devoted to the description of two areas of applications of the above mentioned theories: spectral estimation and model reduction.
First, we describe a new spectral estimation technique based on rational covariance extension theory. Then, we review the basic motivation which lies behind the model reduction theory.
Several methods to reduce the order of a system have been studied in the literature. A method is more preferable than another if it performs not only a good approximation of the original system but also maintains crucial  properties of the original system in the reduction phase. In this talk, we consider systems which are stable and passive. We describe a methodology based on the analytic interpolation theory with a degree constraint for the computation of the reduced order system which preserves the stability and passivity properties of the full order system. Several examples are presented to show the usefulness of the proposed procedure.