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Fig. 4.9. Comparison of controlled Van Der Pol oscillator with its neural model:
( a ) No control has been used for learning ( b ) A random control signal has been
provided to the system during learning
Those results have been obtained with a classical neural architecture with
three inputs, ten hidden neurons and two output neurons. If training is
performed from a training set that has the same size, but that was built
from a regular sampling of the state space and of the feasible control set, the
results are poor (if there is no data preprocessing). That result shows the cru-
cial importance of the selection of the training set. Actually, as emphasized in
Chap. 2, it is important to build the training set according to the probability
distribution of visit on the input space in experimental conditions. That will
be further elaborated when on-line training will be explained. Note the im-
portance of the implemented open-loop control signal to visit the full input
space, especially when the system is pushed on a single attractor (like the
limit cycle of Van der Pol oscillator. In the next chapter, we will come back
to that “ exploration policy ” in the neuro-dynamical programming framework.
The choice of the order of the model is important because it has a direct
impact on the number of parameters to be estimated. It is more critical than
in the linear case. Actually, the problem of selecting the model order is not
fully solved in nonlinear regression theory. In practice, one combines an empir-
ical approach (estimation of the generalization error) and theoretical criteria
that were designed for linear models [Gourieroux 1995]. Moreover, the model
may be validated ex-post using hypothesis testing [Urbani 1993]. Nonadaptive
identification from a representative training set is not especially troublesome
when neural networks with supervised training are used, provided a cautious
methodology, and e cient training algorithms are used.
Similar considerations apply in the framework of adaptive identification,
where one has to use a flow of experimental data in an adaptive way, i.e., as
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