Neural Identification of Controlled Dynamical Systems and Recurrent Networks - Neural Networks: Methodology and Applications - page 241

Remark. An infinite impulse response filter (IIR filter or “recursive filter”)
is characterized by the fact that its response at time k + 1 depends on its
input at time k and on its response at previous times. On the other hand, a
finite impulse response filter (FIR filter or “transverse filter”) is characterized
by the fact that its response at time k + 1 does not depend on its response
at previous instants but solely on the input signal at the same instant and at
previous instants.

In addition, the model of the response of FIR filters to white noise input,
such as

x ( k +1)= b 0 v ( k +1)+ b 1 v ( k )+ ··· + b q v ( k − q +1) ,

are usually called moving average process MA( q ). The natural generalization
of these two models is the auto-regressive moving average model of order ( p,q ),
or ARMA ( p,q ), model.
Although ARMA models enjoy universal approximation property, it is
more e cient to build nonlinear evolution equations to model phenomena
or signals that admit parsimonious nonlinear representations [Tong 1995]. So
NARMA models are introduced with the following regression equation

x ( k +1)= f [ x ( k ) ,...,x ( k − p +1) ,v ( k +1) ,v ( k ) ...,v ( k − q +1)] .

We point out that these models are particular examples of dynamical sys-
tems that have been addressed in previous paragraphs. Their state represen-
tations are obvious but quite voluminous. For instance in the previous order
( p,q ) NARMA model, the state of the system at time k is the vector x ( k ),
which has p + q components, namely,

[ x 1 ( k )= x ( k ) ,..., X P ( k )= x ( k − p +1) , x p +1 ( k )

= v ( k ) ..., x p + q ( k )= v ( k − q +1)] ,

and the state equation is

x 2 ( k +1)= x 1 ( k )
.................................
x p ( k +1)= x p− 1 ( k )
x p +1 ( k +1)= v ( k +1)
x p +2 ( k +1)= x p +1 ( k )
.................................
x p + q ( k +1)= x p + q− 1 ( k )
x 1 ( k +1)= f [ x 1 ( k ) ,..., x p ( k ) ,v ( k +1) , x p +1 ( k ) ..., ) , x p + q ( k )] .

In the same way we considered controlled dynamical systems built from
autonomous dynamical systems by introducing an input, time series the-
ory considers autoregressive models with exogenous inputs which are called