Modeling with Neural Networks: Principles and Model Design Methodology - Neural Networks: Methodology and Applications - page 150

sampled systems: the quantities of interest are measured at discrete times,
which are integer multiples of a sampling period T .
For simplicity, the quantity T will be omitted in equations below: the value
of a variable x at time kT , k positive integer, will be denoted as x ( k ).
Chapter 4 offers a general view of nonlinear dynamic systems. In this chap-
ter, the presentation will be restricted to a cursory introduction to continuous-
state stochastic modeling, which derives directly from the previous discussions
on static modeling. The elements of dynamic modeling that are presented here
are su cient for understanding the methodology of semiphysical modeling,
which is very important for industrial applications.

2.7.1 State-Space Representation and Input-Output
Representation

Dynamic modeling has several specific features, which are not relevant to
static modeling.
The first specific feature is the existence of several representations for the
dynamic model of a given process (see for instance [Kuo 1995] for an introduc-
tion to dynamic systems, and [Kuo 1992] for an introduction to discrete-time
systems). In the following, the modeling of a single-output process is dis-
cussed; its extension to multiple-output systems is relatively straightforward.
A model is said to be a state-space representation if its equations are in the
form:
x ( k )= f ( x ( k − 1) , u ( k − 1) , b 1 ( k

− 1)) state equation

y ( k )= g ( x ( k ) , b 2 ( k )) observation equation or output equation,

where vector x ( k ) is the state vector (whose components are the state vari-
ables), vector u ( k ) is the control input vector, b 1 ( k )and b 2 ( k ) are the vectors
of disturbances, and scalar y ( k ) is the model output. f is a nonlinear vector
function, and g is a nonlinear scalar function. The dimension of the state vec-
tor (i.e., the number of state variables) is called the “order” of the model. The
state variables may be either measured or not measured.
For a single-input process with control input u ( k ), the components of vec-
tor u ( k )maybe u ( k ) and past values of the input control signal: u ( k )=
[ u ( k ) ,u ( k m )] T .

The disturbances have an influence either on the output, or on the state
variables, or on both. As opposed to control inputs, they are not measured.
Therefore, they cannot be inputs of the model, although they do have an
influence on the quantity to be measured. For instance, for an oven, the current
intensity that flows in the heating resistor is a control input; the measurement
noise of the thermocouple is a disturbance that can be modeled, if necessary,
as a sequence of realizations of random variables.
The output may be one of the state variables (an example will be described
in the section “What to do in practice?”.

− 1) ,...,u ( k −