Abstract: The paper presents a new notion of equivalence of non-regular AR-representations, based on the coincidence of the impulsive-smooth behaviours of the underlying systems. The proposed equivalence is characterized by a special case of the usual unimodular equivalence and a restriction of the matrix transformation of full equivalence.
Abstract: In an earlier paper by the present authors, a new family of companion forms associated with a regular polynomial matrix was presented, generalizing similar esults by M. Fiedler who considered the scalar case. This family of companion forms preserves both the finite and infinite elementary divisor structure of the original polynomial matrix, thus all its members can be seen as linearizations of the corresponding polynomial matrix. In this note, its applications on
polynomial matrices with symmetries, which appear in a number of engineering fields, are examined.
Abstract: We propose a new algorithm for the computation of a minimal polynomial basis of the left kernel of a given polynomial matrix F(s). The proposed method exploits the structure of the left null space of generalized Wolovich or Sylvester resultants to compute row polynomial vectors that form a minimal polynomial basis of left kernel of the given polynomial matrix. The entire procedure can be implemented using only orthogonal transformations of constant matrices and results to a minimal basis with orthonormal coefficients.
Abstract: In this paper a new family of companion forms associated to a regular polynomial matrix is presented. Similar results have been presented in a recent paper by M. Fiedler, where the scalar case is considered. It is shown that the new 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. Furthermore, for the special class of self-adjoint polynomial matrices a particular member is shown to be self-adjoint itself.
Abstract: We examine the problem of equivalence of discrete time auto-regressive representations (DTARRs) over a finite time interval. Two DTARRs are defined as fundamentally equivalent (FE) over a finite time interval [0, N] if their solution spaces or behaviours are isomorphic. We generalize the concept of strict equivalence (SE) of matrix pencils to the case of general polynomial matrices and in turn we show that FE of DTARRs implies SE of the underlying polynomial matrices.
Abstract: 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.
Abstract: 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.
Abstract: In this note we study the effect of constant pseudostate feedback on the internal properness of a linear multivariable system, described by an ARMA model. It is shown that the existence of a constant peudostate feedback control law which makes the closed-loop system internally proper is equivalent to the absence of decoupling zeros at infinity of the open-loop system, a well-known result from the theory of descriptor systems.
Abstract: In this paper we investigate the behavior of the discrete time AR (Auto Regressive) representations over a finite time interval, in terms of the finite and infinite spectral structure of the polynomial matrix involved in the AR-equation. A boundary mapping equation and a closed formula for the determination of the solution, in terms of the boundary conditions, are also given.
Abstract: The subject of the present PhD thesis is the study of singular linear discrete time systems. The regular first-order case has been studied by several authors in the past, mainly in an algebraic and geometric analysis level. The main objective of the present study is to extend these results towards three directions. In particular, the analysis of non-regular first-order singular systems, the study of the structure of the solution space (behavior) of auto-regressive (AR) representations and a solution method for auto-regressive moving average (ARMA) representation using the fundamental matrix sequence. The structure of the solution spaces has been directly connected to the algebraic structure of the polynomial matrices describing the corresponding systems while special attention has been given on the infinite algebraic structure of the matrices involved.
