Efstathios N. Antoniou

A permuted factors approach for the linearization of polynomial matrices

In Antoniou and Vologiannidis (Electron J Linear Algebra 11:78–87, 2004; 15:107–114, 2006), a new family of companion forms associated with a regular polynomial matrix T (s) has been presented, using products of permutations of n elementary matrices, generalizing similar results presented in Fiedler (Linear Algebra Its Appl 371:325–331, 2003) where the scalar case was considered. In this paper, extending this “permuted factors” approach, we present a broader family of companion-like linearizations, using products of up to n(n−1)/2 elementary matrices, where n is the degree of the polynomial matrix. Under given conditions, the proposed linearizations can be shown to consist of block entries where the coefficients of the polynomial matrix appear intact. Additionally, we provide a criterion for those linearizations to be block symmetric. We also illustrate several new block symmetric linearizations of the original polynomial matrix T (s), where in some of them the constraint of nonsingularity of the constant term and the coefficient of maximum degree are not a prerequisite.

On the Realization Theory of Polynomial Matrices and the Algebraic Structure of Pure Generalized State Space Systems

On the Realization Theory of Polynomial Matrices and the Algebraic Structure of Pure Generalized State Space Systems We review the realization theory of polynomial (transfer function) matrices via pure generalized state space system models. The concept of an irreducible-at-infinity generalized state space realization of a polynomial matrix is defined and the mechanism of the cancellations of decoupling zeros at infinity is closely examined. The difference between the concepts of irreducibility and minimality of generalized state space realizations of polynomial (transfer function) matrices is pointed out and the associated concepts of dynamic and non-dynamic variables appearing in generalized state space realizations are also examined.

Linearizations of Polynomial Matrices with Symmetries and Their Applications.

In E.N. Antoniou and S. Vologiannidis ( 2004), a new family of companion forms associated to a regular polynomial matrix has been presented generalizing similar results presented by M. Fiedler in M. Fiedler (2003) where the scalar case was considered. This family of companion forms preserves both the finite and infinite elementary divisors structure of the original polynomial matrix, thus all its members can be seen as linearizations of the corresponding polynomial matrix. In this note we examine its applications on polynomial matrices with symmetries which appear in a number of engineering fields

Zero coprime equivalent matrix pencils of a 2 - D polynomial matrix

In this paper we propose a procedure to reduce a 2 - D square polynomial matrix of arbitrary degrees to matrix pencils of the form sX + ZY + A, using zero coprime equivalence. As a further step, we provide the necessary and sufficient condition by which pencils of specific forms, appearing as parametric families, are zero coprime equivalent to a 2 - D regular polynomial matrix. Appropriate examples are provided to illustrate the use of proven results.

On the characterization and parametrization of strong linearizations of polynomial matrices

In the present note, a new characterization of strong linearizations, corresponding to a given regular polynomial matrix, is presented. A linearization of a regular polynomial matrix is a matrix pencil which captures the finite spectral structure of the original matrix, while a strong linearization is one incorporating its structure at infinity along with the finite one. In this respect, linearizations serve as a tool for the study of spectral problems where polynomial matrices are involved. In view of their applications, many linearization techniques have been developed by several authors in the recent years. In this note, a unifying approach is proposed for the construction of strong linearizations aiming to serve as a bridge between approaches already known in the literature.

Notions of equivalence for linear multivariable systems

The present paper is a survey on linear multivariable systems equivalences. We attempt a review of the most significant types of system equivalence having as a starting point matrix transformations preserving certain types of their spectral structure. From a system theoretic point of view, the need for a variety of forms of polynomial matrix equivalences, arises from the fact that different types of spectral invariants give rise to different types of dynamics of the underlying linear system. A historical perspective of the key results and their contributors is also given.

Computation of the general solution of a multivariate polynomial matrix Diophantine equation

The algorithm presented in [21] provides a method for the computation of the general solution of a polynomial matrix Diophantine equation. In this work we extend this algorithm for the n-D PMDE. We present a method to efficiently address the division of multivariate polynomials. The theory is implemented via illustrative examples.

Symbolic Computations on Rings of Rational Functions and Applications in Control Engineering

A collection of algorithms implemented in Mathematica 7.0, freely available over the internet, and capable to manipulate rational functions and solve related control problems using polynomial analysis and design methods is presented. The package provides all the necessary functionality and tools in order to use the theory of

Descriptor systems toolbox : a Mathematica-based package for descriptor systems

it Omega-

Linearizations of polynomial matrices with symmetries and their applications

stable functions, and is expected to provide the necessary framework for the development of several other algorithms that solve specific control problems.