A new notion of equivalence for discrete time AR representations N. P. KARAMPETAKISy*, S. VOLOGIANNIDISy and A. I. G. VARDULAKISy 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. 1. Introduction The problem of equivalence of polynomial matrices has been studied extensively. The primary target of these studies was the preservation of the finite elementary divisors structure of the polynomial matrices involved in the equivalence transformations examined. An equivalence relation for matrix pencils, termed strict equivalence, was initially introduced in Gantmacher (1959) where it was shown to have the property of preserving the finite elementary divisor structure of (strictly) equivalent pencils. Strict equivalence of matrix pencils has been extended to the general polynomial matrix case, in Rosenbrock (1970) by unimodular equiva- lence. However, both strict equivalence and unimodular equivalence can only be applied to matrices of the same dimension. In Pugh and Shelton (1978), extended uni- modular equivalence was introduced as a closed form relation between polynomial matrices of possibly differ- ent dimensions, preserving the finite elementary divisor structure of the polynomial matrices involved. Infinite elementary divisors (IEDs) of matrix pencils were initially defined in Gantmacher (1959) where it was shown that IEDs remain invariant under the transfor- mation of strict equivalence. Furthermore, existing work on strict equivalence of matrix pencils makes no distinc- tion between the preservation of their infinite zeros and infinite elementary divisors structures, since in the case of matrix pencils, the orders of the IEDs are related to the orders of zeros at infinity by the ‘plus one property’ (Vardulakis and Karcanias 1983). Complete equivalence is proposed in Pugh et al. (1987) for matrix pencils of possibly different dimensions, which preserves both finite and infinite zero (elementary divisor) structures of matrix pencils. Hayton et al. (1988) extended the definition of the IEDs to the polynomial matrix case and showed that the IED structure gives a complete description of the total pole-zero structure at infinity of a polynomial matrix and not simply that associated with the zeros at infinity. Thus, although full equivalence presented later in Hayton et al. (1990) preserves the infinite zero structure of polynomial matrices, it does not preserve the IED structure, apart from the special class of polynomial matrices having the same order. Strict equivalence transformation of matrix pencils was generalized for the case of regular, i.e. square and non-singular, polynomial matrices in Vardulakis and Antoniou (2003) where it was shown to preserve both the FED and IED structures of the polynomial matrices involved. This notion of equivalence characterizes strict equivalent regular polynomial matrices through their first-order representations using strict equivalence of matrix pencils (Gantmacher 1959), providing no closed form equivalence relation, something that is its main drawback. Many authors applied the algebraic results of the above studies in order to define equivalence rela- tions between state space and descriptor systems in both the continuous and discrete time cases. In this paper we present a closed form equivalence relation between poly- nomial matrices of possibly different degrees and dimen- sions, which leaves invariants both the finite and infinite elementary divisor structures, thus extending in these terms the notion of strict equivalence of matrix pencils. This work was motivated by the study of discrete time autoregressive representations (DTARR), where the finite and infinite elementary divisor structure of the polynomial matrix describing a DTARR was proved to be of crucial importance (Lewis 1984, Antoniou et al. 1998, Karampetakis 2004). Consider a linear, homogeneous, matrix difference equation AðÞðkÞ¼ 0, k 2½0, N ð1Þ AðÞ¼ A q  q þ A q1  q1 þþ A 0 2 R½ rr ð2Þ where A i 2 R rr , i ¼ 0, 1, ... , q,  denotes the forward shift operator: ðkÞ¼ ðk þ 1Þ and ðkÞ: 0, N ½  !R r is International Journal of Control ISSN 0020–7179 print/ISSN 1366–5820 online # 2004 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/002071709410001703223 INT. J. CONTROL, 15 APRIL 2004, VOL. 77, NO. 6, 584–597 Received 1 October 2003. Revised and accepted 26 March 2004. * Author for correspondence. e-mail: karampet@ math.auth.gr y Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki 54006, Greece.