Factorization and Parametrization

Abstract

In the analysis and design of linear systems, factorization problems arise in a variety of contexts. Herein we focus on the factorization algorithms involved in the H₂ and H_∞ filtering problems, known in the literature as spectral factorization and J-spectral factorization, respectively. By duality, the corresponding techniques for control can be easily derived. For the introduction to factorization methods, the extension of a few concepts of the linear systems theory is necessary. In particular, we will dwell upon the concept of adjoint system, already introduced in Sect. 2.5, and that of symplectic pencil. Taking advantage of all these items, the important issue of the parametrization of stabilizing controllers will be treated in the last part of the chapter. The parametrization problem will be tackled with reference to both the polynomial and rational descriptions of periodic systems.