**2. Neural networks and static modelling**

We are introducing the term of static modelling of systems. Static modelling is used to model the time independent properties of systems which implies that the systems behaviour remains relatively unchanged within the time frame important for the application. (Fig. 1). In this category we can understand also the systems which do change their reaction on stimulus, but this variability is measurable and relatively stable in the given time period. We regard the system as static when its reaction on stimulus is stable and most of all repeatable – in some sense - static.

The formal description of static system (Fig. 1) is given in (1)

$$Y\_m(X\_m, \mathbf{t}) = f\left(X\_m, P\_m, \mathbf{t}\right) \tag{1}$$

Where *Ym* is the m - dimensional output vector,

*Xn* is the n – dimensional input – stimulus vector, *Pu* is the system parameters vector, *t* is the time.

In order to regard the system as static both the function *f* and the parameters vector *Pu* do not change in time.

Fig. 1. The formal description of static system.

The first sub-chapter starts with an introduction to the terminology used for neural networks. The terminology is essential for adequate understanding of further reading.

The section entitled "Some critical aspects" summarizes the basic understanding of the topic

The users who want to use neural network tools should be aware of the problems posed by the input and output limitations. These limitations are often the cause of bad modelling results. A detailed analysis of the neural network input and output considerations and the

In practice the neural network modelling of systems that operate on a wide range of values represents a serious problem. Two methods are proposed for the approximation of wide

A very important topic of training stability follows. It defines the magnitude of diversity detected during the network training and the results are to be studied carefully in the course

At the end of the chapter the general design steps for a specific neural network modelling

We are introducing the term of static modelling of systems. Static modelling is used to model the time independent properties of systems which implies that the systems behaviour remains relatively unchanged within the time frame important for the application. (Fig. 1). In this category we can understand also the systems which do change their reaction on stimulus, but this variability is measurable and relatively stable in the given time period. We regard the system as static when its reaction on stimulus is stable and most of all repeatable

In order to regard the system as static both the function *f* and the parameters vector *Pu* do

*<sup>f</sup>*(*<sup>X</sup>* ,*P*,*t*) *<sup>n</sup> <sup>u</sup> Xn Ym*

*Ym(Xn, t) = f (Xn, Pu, t)* (1)

and shows some of the errors in formulations that are so often made.

errors that may be produced by these procedures are given.

range functions.

task are given .

– in some sense - static.

*t* is the time.

not change in time.

of any serious data modelling attempt.

**2. Neural networks and static modelling** 

Where *Ym* is the m - dimensional output vector, *Xn* is the n – dimensional input – stimulus vector,

Fig. 1. The formal description of static system.

*Pu* is the system parameters vector,

The formal description of static system (Fig. 1) is given in (1)

Under the formal concept of static system we can also imply a somewhat narrower definition as described in (1). Here the system input – output relationship does not include the time component (2).

$$Y\_m \ (X\_n) = f(X\_n \ P\_u) \tag{2}$$

Although this kind of representation does not seem to be practical, it addresses a very large group of practical problems where the nonlinear characteristic of a modelled system is corrected and accounted for (various calibrations and re-calibrations of measurement systems).

Another understanding of static modelling refers to the relative speed (time constant) of the system compared to the model. Such is the case where the model formed by the neural network (or any other modelling technique) runs many times faster than does the original process which is corrected by the model1.

We are referring to the static modelling when the relation (3) holds true.

$$
\tau\_m \ll \tau\_s \tag{3}
$$

Where *m* represents the time constant of the model, and *<sup>s</sup>*represents the time constant of the observed system. Due to the large difference in the time constants, the operation of the model can be regarded as instantaneous.

The main reason to introduce the neural networks to the static modelling is that we often do not know the function *f* (1,2) analytically but we have the chance to perform the direct or indirect measurements of the system performance. Measured points are the entry point to the neural network which builds the model through the process of learning.
