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Thus, the designer of a state-space model seeks approximations of func-
tions f and g , through training from sequences of inputs, of outputs and
possibly of state variables if the latter are measured.
A model is in input-output representation if its equations are in the form
y ( k )= h ( y ( k − 1) ,...,y ( k −n ) , u ( k − 1) ,..., u ( k −m ) , b ( k − 1) ,...,
b ( k −p )) ,
where h is a nonlinear function, n is the order of the model, m and p are
two positive integer constants, u ( k ) is the vector of input control signals, b ( k )
is the vector of disturbances. Input-output representations are special forms
of state-space representations, where the components of the state vector are
[ y ( k
n )].
In linear modeling, state-space representations and input-output represen-
tations are equivalent: one chooses the representation that is most convenient
in view of the purpose that the model is intended to serve. By contrast, in
nonlinear modeling, state-space representations are more general and more
parsimonious than input-output models [Levin et al. 1993], as will be illus-
trated below on a real application; however, the design of a state-space model
may be slightly more di cult than that of an input-output model, since two
functions f and g must be approximated, while input-output models require
the approximation of a single function h .
Once a choice has been made between state-space and input-output rep-
resentation, an assumption must be made as to the influence of noise on the
process. That is a basic fact that is often overlooked in the neural network
literature, whereas it is common knowledge in linear dynamic modeling, as is
shown in Chap. 4. In the present chapter, we show that the assumption on
the noise has a deep influence on the training algorithm that must be used, on
the structure of the model that must be implemented, and on its subsequent
mode of operation. In the next section, the main assumptions on noise are
discussed, and the resulting constraints on the training of the model, on its
structure and on its operation are explained.
2.7.2 Assumptions on Noise and Their Consequences on the
Structure, the Training and the Operation of the Model
1) ,y ( k
2) ,...y ( k
In the present section, various assumptions on the influence of noise on the
process are considered. We first discuss the assumptions and their conse-
quences on the structure, training and operation of input-output models, then
the consequences of the assumptions on state-space models.
2.7.2.1 Input-Output Representations
State Noise Assumption (Input-Output Representation)
We assume that the model can be appropriately described, in the desired
validity domain, by a representation of the form
y p ( k )= ψ ( y p ( k
1) ,...,y p ( k
n ) , u ( k
1) ,..., u ( k
m )) + b ( k ) ,
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