Fundamental equivalence of discrete-time AR representations

A. I. G. Vardulakis, E. Antoniou
2003 International Journal of Control  
We examine the problem of equivalence of discrete time auto-regressive (AR) representations. Two AR representations are defined as fundamentally equivalent if their solution spaces or behaviors are isomorphic. Starting from the fact that the behavior of an AR representation, when considered over a finite time interval, depends on the algebraic structure of both the finite and the infinite elementary divisors of the underlying polynomial matrix we examine closely this structure and show that it
more » ... e and show that it can be easily exposed through the corresponding structure of a block companion matrix pencil which can be easily constructed from the coefficients of the original matrix. As a consequence the proposed block companion matrix pencil constitutes the natural first-order fundamentally equivalent representation (realization) of such an AR representation. As a further consequence and generalization we show that two AR representations described by polynomial matrices of possibly different degrees and dimensions that give rise to fundamentally equivalent first-order representations are fundamentally equivalent.
doi:10.1080/0020717031000123607 fatcat:v3a7hyr7efdfpn5pw7x43cqtxm