System Identification Technique and Neural Networks for Material Lifetime Assessment Application - Complex System Modelling and Control Through Intelligent Soft Computations - page 767

where u(n) and y(n), respectively, state the input and output of the model at discrete
time n; u(n), y(n) .

Moreover, d y and d u are the output-memory and input-memory orders. d y rep-
resents the number of lagged output values, which is often referred to as the order of
the model, d u represents the number of lagged input values (d u , d y ≥

ϵ <

1 and d u ≤ d y ).

The vectors y(n) and u(n), therefore, form the output and input regressors,

respectively.
The NARX model is commonly trained using two basic modes, namely:

1. Parallel (P) Mode

Using this mode, the output regressor utilized the estimated outputs which are
fed back to the regressor.

Þ ¼f y ^ ð n þ 1 ^ y ð n Þ ; ...; ^ y ð n d y þ 1 Þ; u ð n Þ; u ð n 1 Þ ; ...; u ð n d u þ 1 Þ ð 7 Þ

2. Series-Parallel (SP) Mode

Using this mode, the output regressor utilized the actual output values.

Þ ¼ ^ fy

y ^ ð n þ 1 ð n Þ ; ...; y ð n d y þ 1 Þ; u ð n Þ; u ð n 1 Þ ; ...; u ð n d u þ 1 Þ ð 8 Þ

It is worth to note that standard feed-forward architecture trained with back-
propagation (BP) technique can be used directly in the NARX mode of SP. In
addition, various learning algorithms are also widely applicable. A form of regu-
larization may also be employed because the additive measurement errors,

ϵ n ,

2 , can be also present.
Figure 4 illustrates the NARX with input and output tapped delay lines (TDL), in
parallel and series-parallel architectures (Neural Network Toolbox User

which are zero-mean Gaussian variables with Var[

ϵ n ]=

˃

' s Guide

1992 ).

Fig. 4 Parallel and series-parallel architectures of NARX network