Forward, backward, and symmetric solutions of discrete ARMA representations

The main objective of this paper is to determine a closed formula for the forward, backward, and symmetric solution of a general discrete-time Autoregressive Moving Average representation. The importance of this formula is that it is easily implemented in a computer algorithm and gives rise to the solution of analysis, synthesis, and design problems.     

Inverses of Multivariable Polynomial Matrices by Discrete Fourier Transforms

The problem of the fast computation of the Moore–Penrose and Drazin inverse of a multi-variable polynomial matrix is addressed. The algorithms proposed, use evaluation-interpolation techniques and the Fast Fourier transform. They proved to be faster than other known algorithms. The efficiency of the algorithms is illustrated via randomly generated examples.     

A new notion of equivalence for discrete time AR representations

We present a new equivalence transformation termed divisor equivalence, that has the property of preserving both the finite and the infinite elementary divisor structures of a square non-singular polynomial matrix. This equivalence relation extends the known notion of strict equivalence, which dealt only with matrix pencils, to the general polynomial matrix case. It is proved that divisor equivalence characterizes in a closed form relation the equivalence classes of polynomial matrices that give rise to fundamentally equivalent discrete time auto-regressive representations.     

On the Newton bivariate polynomial interpolation with applications

The main purpose of this work is to provide recursive algorithms for the computation of the Newton interpolation polynomial of a given two-variable function. The special case where the interpolation polynomial has known upper bounds on the degree of each indeterminate is studied and applied to the computation of the inverse of a two-variable polynomial matrix.     

Special issue on the use of computer algebra systems for computer aided control system design

The importance of the continuing and growing need in the systems and control community for reliable algorithms and robust numerical software for increasingly challenging applications is well known and has already been reported elsewhere (IEEE Control Systems Magazine, Vol. 24, Issue 1). However, we have all had the experience of working on a mathematical project where an increased number of symbolic manipulations was needed. In a simple case, the required computation might have been to compute the Laplace transform or the inverse Laplace transform of a function, or to find the transfer function matrix for a given system topology where parameters are included. In a more demanding situation the required computation might have been to find the parametric family of solutions of a polynomial matrix Diophantine equation resulting from a variety of control problems such as those associated with stabilization, decoupling, model matching, tracking and regulation, or to compute the Smith McMillan form of a rational transfer function matrix in order to obtain a better insight into a number of structural properties of a system. The desire to use a computer to perform long and tedious mathematical computations such as the above led to the establishment of a new area of research whose main objective is the development: (a) of systems (software and hardware) for symbolic mathematical computations, and (b) of efficient symbolic algorithms for the solution of mathematically formulated problems. This new subject area is referred to by a variety of terms such as symbolic computations, computer algebra, algebraic algorithms to name a few. During the last four decades this subject area has accomplished important steps and it is still continuing its evolution process.     

Generalized inverses of two-variable polynomial matrices and applications

The main contribution of this paper is to present (a) an algorithm for the computation of the generalized inverse of a not necessarily square two-variable polynomial matrix and (b) some applications of the proposed algorithm to the solution of Diophantine equations.This work is supported by the Greek General Secretariat of Industry, Research and Technology.