Deterministic ARMA Models

  • J. W. Nieuwenhuis
  • J. C. Willems
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 83)


This paper deals with the ubiquitous class of linear time-invariant finite dimensional systems. In [1] it has been shown that this is precisely the class of AR systems, that is, the dynamical systems whose behavior is governed by a finite set of autoregressive relations. We emphasize that the term autoregressive should here be understood in a purely deterministic sense.

This attractive representation result leads to the following questions:
  1. i)

    what should we mean by a — deterministic — MA (moving average) system and by an ARMA (autoregressive moving average) system?

  2. ii)

    what class of systems admi an MA and an ARMA representation?

We will see that whereas AR representations always exist, it are only the reachable systems which admit an MA representation.

The interplay between AR and MA representation turns out to be very fruitful indeed, and we will sketch a few facets of it in this paper: their relation with left and right coprime factorizations of the transfer functions; their immediate relation with duality; and finally their potential of inducing prime decompositions of systems.

It is worthwile to emphasize that models of systems universally involve auxil-iary variables and as such models obtained from first principles will essentially always be of the ARMA type. As such the reduction to AR, MA, Input/Output or State Space forms is in our opinion one of the important problems of our field.

It is easy to see that AR and MA representations are not unique. This fact in-duces natural equivalence relations on the class of polynomial matrices with a fixed number of columns or rows. We specify what this equivalence means, in terms of the polynomial matrices involved and we give some (initial) results on invariants and canonical forms.


Auxiliary Variable Polynomial Matrix Full Column Rank Polynomial Matrice Polynomial Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Science+Business Media Dordrecht 1986

Authors and Affiliations

  • J. W. Nieuwenhuis
    • 1
  • J. C. Willems
    • 1
  1. 1.Econometrics Institute Mathematics InstituteUniversity of GroningenAV GroningenThe Netherlands

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