In this paper, the authors deal with rational transfer function linear systems (ARMAX systems), show their algebraic and topological structure and algorithms to carry out the fitting of a model and their asymptotic properties.
Section 1 is introduction. In section 2, they summarize several results concerning the matrix transfer functions, the rank of Hankel matrix, state-space forms of the systems, some topological and algebraic properties of the spaces of matrices of rational functions and two problems arising in estimating a true ARMAX structure by examining a subspace related to matrix transfer functions.
Section 3 deals with the algorithms for fitting an ARMAX model to a system. This means they propose a method to obtain the orders of a model and the estimates of the system parameters. For this purpose, they propose four stages. Stage I is to calculate the Toeplitz regression. Stages II and III are concerned with the estimation of the orders of the autoregressive and the moving average parts and recursive calculation of the estimates of the system parameters. They present an algorithm which is recursive in the order parameters. Stage IV is an asymptotically efficient estimation of the system parameters. They mention alternative procedures and show some comments.
In section 4, some simulation results are shown, especially, for stages II and III. And also they show the results of analysis of real data and mention the use of the Kalman filter and coordinate neighbourhoods.
In section 5, they show some theorems concerning the algorithms. Here they give the rigorous representation of the system. They show the convergence of the estimators and the limits of the estimators of the orders, which are appearing in the algorithms.