ABSTRACT
A direct extension of linear dynamic models is the Voltenn series representation [92]. The Volterra representation is very general. In practice, however, a finite trnncation of the series must be used, and a discrete approximation of the series made. For a SISO system, a Volterra model can be given by
114 CHAPTER 5. NON-LINEAR DYNAMIC STRUCTURES
5.1. NON-LINEAR TIME-SERIES MODELS 115
y(k) = f(u (k-d) 1 ••• , U (k-d-nB) 1 y(k-1) 1 ••• 1 y(k-nA) 1 W) (5.2) In the N ARX structure the input consists of past inputs and outputs of the process:
The structure of the mapping f between the inputs and the output is not determined. If no a priori information about the structure of the process is available, it is common to choose some black-box structure: power series, sigmoid neural networks, or 0-order Sugeno fuzzy system (among many others, see Chapter 4). In practice, process modelling using NOE, NARX and NARMAX structures can give accurate predictions on a fixed data set. Most importantly, it is possible to model a wide class of non-linearities. If some non-linear black-box structure is chosen for the static function f, practically all reasonable dynamic functions can be approximated (provided that the input data windows are long enough, and the size (number of parameters) of the black-box model is sufficiently large). The approach is simple, as it extends the linear dynamic time-series structures to non-linear combinations of the inputs. If the mapping f is a linear one, ARX, OE and ARMAX structures result (see Chapter 3).
