From time series to linear system—Part I. Finite dimensional linear time invariant systems

Abstract

In the first part of this paper the definition of a dynamical system as simply consisting of a family of time series will be developed. In this context the notions of linearity, time invariance, and finite dimensionality will be introduced. It will be shown that a given family of time series may be represented by a system of (AR) equations: R i w( t + l) + R l − 1 w( t + l − 1) + … + R 0w( t) = 0, or, equivalently, by a finite dimensional linear time invariant system: x( t + 1) = Ax( t) + Bu( t); y( t) = Cx( t) + Du( t); w = (u, y), if and only if this family is linear, shift invariant and complete (or, as is equivalent, closed in the topology of pointwise convergence). This yields a very high level and elegant set of axioms which characterize these familiar objects. It is emphasized, however, that no a priori choice is made as to which components of w are inputs and which are outputs. Such a separation always exists in any specific linear time invariant model. Starting from these definitions, the structural indices of such systems are introduced and it is shown how an (AR) representation of a system having a given behaviour can be constructed. These results will be used in a modelling context in Part II of the paper.