Parametrization of minimal spectral factors of discrete-time rational spectral densities

In this paper, the problem of providing a complete parametrization of the minimal spectral factors of a discrete-time rational spectral density is considered. The desired parametrization, given in terms of the all-pass divisors of a certain all-pass function, is established in the most general setting: after several partial results, mostly in the continuous-time case, this is indeed the first complete parametrization obtained without resorting to any facilitating assumption. This result provides a positive answer to a conjecture raised in 2016

Novel Results on the Factorization and Estimation of Spectral Densities

This dissertation is divided into two main parts. The first part is concerned with one of the most classical and central problems in Systems and Control Theory, namely the factorization of rational matrix-valued spectral densities, commonly known as the spectral factorization problem. Spectral factorization is a fundamental tool for the solution of a variety of problems involving second-order statistics and quadratic cost functions in control, estimation, signal processing and communications. It can be thought of as the frequency-domain counterpart of the ubiquitous Algebraic Riccati Equation and it is intimately connected with the celebrated Kálmán-Yakubovich-Popov Lemma and, therefore, to passivity theory. Here, we provide a rather in-depth and comprehensive analysis of this problem in the discrete-time setting, a scenario which is becoming increasingly pervasive in control applications. The starting point in our analysis is a general spectral factorization result in the same vein of Dante C. Youla. Building on this fundamental result, we then investigate some key issues related to minimality and parametrization of minimal spectral factors of a given spectral density. To conclude, we show how to extend some of the ideas and results to the more general indefinite or J-spectral factorization problem, a technique of paramount importance in robust control and estimation theory. In the second part of the dissertation, we consider the problem of estimating a spectral density from a finite set of measurements. Following the Byrnes-Georgiou-Lindquist THREE (Tunable High REsolution Estimation) paradigm, we look at spectral estimation as an optimization problem subjected to a generalized moment constraint. In this framework, we examine the global convergence of an efficient algorithm for the estimation of scalar spectral densities that hinges on the Kullback-Leibler criterion. We then move to the multivariate setting by addressing the delicate issue of existence of solutions to a parametric spectral estimation problem. Eventually, we study the geometry of the space of spectral densities by revisiting two natural distances defined in cones for the case of rational spectra. These new distances are used to formulate a "robust" version of THREE-like spectral estimation

Essays on Identification of Structural VARMA models

This research has been possible thanks to the financial support from the scholarship MCIN/BES-2017-081997 from the Spanish Agency of Science and Innovation (Ministerio de Ciencia e Innovación).Programa de Doctorado en Economía por la Universidad Carlos III de MadridPresidente: Juan José Dolado Lobregad.- Secretario: Dante Amengual.- Vocal: Geert Mester