Nicholas P. Karampetakis

Computation of the Generalized Inverse of a Polynomial Matrix and Applications

Abstract The computation of the generalized inverse of a constant matrix is utilized in finding the generalized inverse and its Laurent expansion of a nonregular polynomial matrix. The proposed algorithm constitutes a generalization of the algorithm proposed by Fragulis et al. for regular polynomial matrices and gives rise to numerous applications in multivariable system analysis.

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.

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.

On the solution and impulsive behaviour of polynomial matrix descriptions of free linear multivariable systems

In this note we examine the solution and the impulsive behaviour of autonomous linear multivariable systems whose pseudo-state beta(t) obeys a linear matrix differential equation A(rho)beta(t) = 0 where A(rho) is a polynomial matrix in the differential operator rho:=d/dt. We thus generalize to the general polynomial matrix case some results obtained by Verghese and colleagues which regard the impulsive behaviour of the generalized state vector x(t) of input free generalized state space systems.

On the discretization of singular systems

This note proposes two new discretization methods. The proposed sampled systemsare described in terms of the Markov parameters of the system and therefore theproposed methods are easily implemented. The methodology we use is a zero-order holddiscretization for the input and first-order approximation of its derivatives.Keywords: discretization; singular systems; implicit systems; sampled systems; zero-orderhold; first-order hold; Markov parameters.

On the Newton multivariate polynomial interpolation with applications

The main purpose of this work is to provide an optimized version of the Newton divided-difference algorithm presented by [11] for the polynomial interpolation problem. This optimized algorithm is used for the computation of the determinant of a two-variable polynomial matrix.

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.

Forward, backward and symmetric solutions of discrete time 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.

Computation of the transfer function matrix and its Laurent expansion of generalized two-dimensional systems

An algorithm is developed for the computation of the transfer function matrix of a two-dimensional system, which is given in its generalized form. The algorithm is a recursion in terms of the original system matrix and does not require the inversion of a two-variable polynomial matrix. An algorithm for the evaluation of the Laurent expansion of the inverse of a two-variable polynomial matrix is also presented.

On the computation of the inverse of a two-variable polynomial matrix by interpolation

Two interpolation algorithms are presented for the computation of the inverse of a two variable polynomial matrix. The first interpolation algorithm, is based on the Lagrange interpolation method that matches pre-assigned data of the determinant and the adjoint of a two-variable polynomial matrix, on a set of points on several circles centered at the origin. The second interpolation algorithm is using discrete fourier transforms (DFT) techniques or better fast fourier transforms which are very efficient algorithms available both in software and hardware and that they are greatly benefitted by the existence of a parallel environment (through symmetric multiprocessing or other techniques). The complexity of both algorithms is discussed and illustrated examples are given. The DFT-algorithm is implemented in the Mathematica programming language and tested in comparison to the respective built-in function of Mathematica.