Algebraic and polynomial computational methods in control theory

Polynomial matrix theory is very important to many automatic control related pro- blems. This thesis focuses on basic theoretical problems of the polynomial point of view of automatic control, but also on the development of algorithms dealing with polynomial matrices of one or more variables. A new equivalence between poly- nomial matrices was introduced, that generalizes the well known strict equivalence between matrix pencils, having as invariants the finite and infinite elementary di- visor structure of the polynomial matrices involved. A new family of companion forms has also been presented, having a particularly simple structure, whose mem- bers can be written as a product of certain elementary sparse matrices. In this fa- mily not only the well known first and second companion forms are included, but also other companion forms with very interesting properties. New algorithms for the calculation of the Moore-Penrose and Drazin inverses of polynomial matrices of one or more variables have been introduced, combining evaluation/interpolation techniques and the Fast Fourier Transform (FFT). Also a new algorithm concer- ning the calculation of a minimal polynomial basis of the kernel of a polynomial matrix has been proved, that uses successive Sylvester orWolovish resultants. The algorithms have been analyzed regarding their speed and numerical robustness and proved to be more efficient than the others known to the literature