On the solution space of discrete time AR-representations over a finite time horizon

Abstract

The main purpose of this work is to determine the forward and backward solution space of a nonregular discrete time AR-representation i.e. A(σ )ξ(k) = 0, in a finite time horizon where A(σ ) is a polynomial matrix and σ is the forward shift operator. The construction of the behavior is based on the structural invariants of the polynomial matrix that describes the AR-representation i.e. the finite and infinite elementary divisors and the right and left minimal indices of A(σ Let R, C denote the fields of real and complex numbers respectively, R[σ ] the ring of polynomials with real coefficients and R(σ ) the field of rational functions. By R(σ ) p×m , R pr (σ ) p×m and R[σ ] p×m we denote the sets of p × m rational, proper rational and polynomial matrices respectively with real coefficients and indeterminate σ . Consider a system of linear homogeneous difference and algebraic equations described in matrix form by A(σ )ξ(k) = 0,(1) *

Construction of Algebraic and Difference Equations with a Prescribed Solution Space

“…r×r and det A(σ) = 0, the algebraic structure of A(σ) and by extension the solution space of (2) are connected with additional invariants due to the left and right null space of A(σ) (see Karampetakis, 2004;Praagman, 1991).…”

“…If the Jordan pairs of A(σ) are not given beforehand, a method for constructing them can be found in the works of Gohberg et al (2009) and Karampetakis (2004). In the following sections we study the inverse problem, that is: Given a specific forward or backward behavior, how to construct a polynomial matrix A(σ) and its corresponding homogenous system A(σ)β(k) = 0 that will satisfy the given behavior.…”

“…This gives rise to a third solution vector for the system, linearly independent of β 1 (k) and β 2 (k). Using the method proposed by Karampetakis (2004), we find that the third solution is …”

“…The number q is often called the lag of studied by various authors (Antoniou et al, 1998;Antsaklis and Michel, 2006;Gohberg et al, 2009;Karampetakis, 2004). A method for constructing the forward (resp.…”

“…infinite) elementary divisor structure of A(σ), was presented by Gohberg et al (2009) (resp. Antoniou et al, 1998), whereas the results were extended for non-regular systems by Karampetakis (2004). The algebraic structure of polynomial matrices was studied by Bernstein (2009), Gantmacher (1959), Gohberg et al (2009), Hayton et al (1988), Kaczorek (2007), Karampetakis and Vologiannidis (2003), Karampetakis et al (2004), Praagman (1991), Zaballa and Tisseur (2012), and in the references cited therein.…”